Transcript Lecture 22

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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Light is an electromagnetic wave
An electromagnetic wave is a traveling wave that has time-varying electric and magnetic
Fields that are perpendicular to each other and the direction of propagation z.
Fig 9.1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Ex = Eo cos(tkz + )
Ex = electric field along x at position z at time t,
k = propagation constant, or wavenumber = 2/
 = wavelength
 = angular frequency
Eo = amplitude of the wave
 is a phase constant which accounts for the fact that at t =
0 and z = 0 Ex may or may not necessarily be zero
depending on the choice of origin.
(tkz + ) =  = phase of the wave .
This equation describes a monochromatic plane wave of
infinite extent traveling in the positive z direction.z
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Wavefront
A surface over which the phase of a wave is constant is
referred to as a wavefront. A wavefront of a plane wave is a
plane perpendicular to the direction of propagation.
The interaction of a light wave with a non-conducting matter
(conductivity,  = 0) uses the electric field component Ex
rather than By.
The optical field refers to the electric field Ex.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
A plane EM wave traveling along z, has the same Ex (or By) at any point in a given xy plane
All electric field vectors in a given xy plane are therefore in phase. The xy planes are of
Infinite extent in the x and y directions.
Fig 9.2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Exponential Notation
Recall that
cos = Re[exp(j)]
where Re refers to the real part. We then need to take the real
part of any complex result at the end of calculations. Thus,
Ex(z,t) = Re[Eoexp(j)expj(tkz)]
or
Ex(z,t) = Re[Ecexpj(tkz)]
where Ec = Eoexp(jo) is a complex number that represents the
amplitude of the wave and includes the constant phase
information o.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Wavevector
We indicate the direction of propagation with a vector k, called
the wavevector, whose magnitude is the propagation constant,
k = 2/. It is clear that k is perpendicular to constant phase
planes.
When the electromagnetic (EM) wave is propagating along
some arbitrary direction k, then the electric field E(r,t) at a
point r on a plane perpendicular to k is
E (r,t) = Eocos(tkr + )
If propagation is along z, kr becomes kz. In general, if k has
components kx, ky and kz along x, y and z, then from the
definition of the dot product, kr = kxx + kyy + kzz.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Wavevector
A traveling plane EM wave along a direction k.
Fig 9.3
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Phase Velocity
The time and space evolution of a given phase , for example
that corresponding to a maximum field is described by
 = tkz +  = constant
During a time interval t, this constant phase (and hence the
maximum field) moves a distance z. The phase velocity of
this wave is therefore z/t. The phase velocity v is
dz 
v
  
dt k
where  is the frequency ( = 2π).
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Phase change over a distance z
The phase difference between two points separated
by z is simply kz since t is the same for each
point.
If this phase difference is 0 or multiples of 2 then
the two points are in phase. Thus, the phase
difference  can be expressed as kz or 2z/.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index
When an EM wave is traveling in a dielectric
medium, the oscillating electric field polarizes the
molecules of the medium at the frequency of the
wave.
The stronger is the interaction between the field
and the dipoles, the slower is the propagation of
the wave.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Phase Velocity and r
The relative permittivity r measures the ease with which the
medium becomes polarized and hence it indicates the extent
of interaction between the field and the induced dipoles.
For an EM wave traveling in a nonmagnetic dielectric
medium of relative permittivity r, the phase velocity v is
given by
v
1
 r o  o
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index
If the frequency  is in the optical frequency range then r
will be due to electronic polarization as ionic polarization will
be too sluggish to respond to the field. However, at the
infrared frequencies or below, the relative permittivity also
includes a significant contribution from ionic polarization and
the phase velocity is slower. For an EM wave traveling in free
space, r = 1 and vvacuum = 1/[oo] = c = 3108 m s–1, the
velocity of light in vacuum. The ratio of the speed of light in
free space to its speed in a medium is called the refractive
index n of the medium,
c
n   r
v
Intuitively makes sense that light propagates more slowly in a denser medium
which has a higher refractive index
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index and Wavevector
Suppose that in free space ko is the wavevector (ko = 2π/)
and o is the wavelength, then the wavevector k in the
medium will be nko and the wavelength  will be o/n.
Indeed, we can also define the refractive index in terms of the
wavevector k in the medium with respect to that in vacuum
ko ,
k
n
ko
In non-crystalline materials such as glasses and liquids,
the material structure is the same in all directions and n
does not depend on the direction. The refractive index is
then isotropic.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index and Isotropy
Typically noncrystalline solids such as glasses and liquids, and cubic
crystals are optically isotropic; they possess only one refractive
index for all directions.
Crystals, in general, have nonisotropic, or anisotropic, properties.
Depending on crystal structure, the relative permittivity r is different
along different crystal directions.
In general, the refractive index n, seen by a propogating EM
wave in a crystal will depend on the value of r along the direction of
the oscillating electric field (i.e., along the direction of polarization).
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Example: Relative permittivity and refractive
index
Relative permittivity r or the dielectric constant of
materials is frequency dependent and further it depends
on crystallographic direction since it is easier to polarize
the medium along certain directions in the crystal. Glass
has no crystal structure, it is amorphous. The relative
permittivity is therefore isotropic but nonetheless
frequency dependent.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The relationship n = (r) between the refractive index n
and r must be applied at the same frequency for both n
and r.
At low frequencies all polarization mechanisms present
can contribute to r whereas at optical frequencies only
the electronic polarization can respond to the oscillating
field.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Electronic Polarization, r and n
Fig 9.4
Electronic polarization of an atom. In the presence of a field in the +x direction, the
electrons are displaced in the –x direction (from O), and the restoring force is in the +x
direction.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Induced Polarization
First, consider applying a dc field. The restoring force Fr
acting on the electron shell is x. In equilibrium, the net
force on the negative charge shell is zero or ZeE + (x) = 0
from which x is known. Therefore the magnitude of the
induced electronic dipole moment is
pinduced = (Ze)x = (Z2e2/)E
Suppose we suddenly remove the applied electric field
polarizing the atom. The equation of motion of the negative
charge center is then (force = mass  acceleration):
–x = Zmed2x/dt2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Resonant or Natural Frequency
Always acts to pull the electrons toward the nucleus O
Restoring force = –x = Zmed2x/dt2
Solving this Diff. Eqn., we can show that:
The displacement at any time is a simple harmonic motion
x(t) = xocos(ot)
where the angular frequency of oscillation o is
  

 o  

Zme 

1 /2
Fig 9.1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
This is the oscillation frequency of the center of mass of the
electron cloud about the nucleus and xo is the displacement
before the removal of the field. After the removal of the field,
the electronic charge cloud executes simple harmonic motion
about the nucleus with a natural frequency o; o is also
called the resonance frequency.
The oscillations of course die out with time because there is
an inevitable loss of energy from an oscillating charge cloud.
An oscillating electron is like an oscillating current and loses
energy by radiating electromagnetic waves; all accelerating
charges emit radiation.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Forced Oscillations by the Optical Field
Consider now the presence of an
oscillating electric field due an
electromagnetic wave passing through the
location of this atom.
The applied field oscillates harmonically in
the +x and x directions, that is E =
Eoexp(jt).
Newton’s second law for Ze electrons with
mass Zme driven by E is given by,
2
d x
Zme 2   ZeEo exp( jt )  x
dt
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Displacement of Electrons by the Oscillating Optical Field
The solution of this equation gives the instantaneous
displacement x(t) of the center of mass of electrons from the
nucleus (C from O),
eEo exp( jt )
x  x(t )  
2
2
me (o   )
The induced electronic dipole moment is then simply given
by pinduced = (Ze)x.
The electronic polarizability e is the induced dipole moment
per unit electric field,
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Induced Polarization and Electronic Polarizability
pinduced
Ze 2
e 

2
2
E
me (o   )
The displacement x and hence electronic polarizability e
increase as  increases. Both become very large when 
approaches the natural frequency o.
The simplest (and a very “rough”) relationship between the
relative permittivity r and polarizability e is
 r  1
N
o
e
where N is the number of atoms per unit volume.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Dispersion Relation
where N is the number of atoms per unit volume. Given that the
refractive index n is related to r by n2 = r, it is clear that n must
be frequency dependent, i.e.
2 

NZe
1
2

n  1  

 2
2

m



 o e  o
We can also express this in terms of the wavelength. If o =
2c/o is the resonance wavelength, then
2
2 
2


NZe


2
o




n  1  

 2

2

m

2

c




 o e 
o
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Dispersion Relation
The relationship between n and the frequency , or wavelength
, is called the dispersion relation.
n will always be wavelength dependent and will exhibit a
substantial increase as the frequency increases towards a natural
frequency of the polarization mechanism.
We considered the electronic polarization of an isolated atom
with a well-defined natural frequency o. In the crystal,
however, the atoms interact and further we also have to
consider the valence electrons in the bonds. The overall result is
that n is a complicated function of the frequency or the
wavelength.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Velocity and Group Index
There are no perfect monochromatic waves in practice.
We have to consider the way in which a group of waves
differing slightly in wavelength will travel along the zdirection.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Velocity and Group Index
When two perfectly harmonic waves of frequencies 
and  +  and wavevectors kk and k + k interfere, they
generate a wave packet which contains an oscillating field at
the mean frequency  that is amplitude modulated by a
slowly varying field of frequency . The maximum
amplitude moves with a wavevector k and thus with a group
velocity that is given by
d
vg 
dk
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Two slightly different wavelength waves traveling in the same direction result in a wave
packet that has an amplitude variation that travels at the group velocity.
Fig 9.5
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The group velocity therefore defines the speed with which
energy or information is propagated.
Since  = vk and the phase velocity v = c/n, the group
velocity in a medium can be readily evaluated. Suppose
that v depends on the wavelength or k by virtue of n being
a function of the wavelength as in the case for glasses.
Then,
 c 2 
 
  vk  

n( ) 
 
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Velocity and Group Index
where n = n() is a function of the wavelength. The group
velocity vg in a medium is approximately given by,
d
c
v g (medium) 

dk n   dn
d
This can be written as
c
v g (medium) 
Ng
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Index
dn
Ng  n  
d
is defined as the group index of the medium.
In general, for many materials the refractive index n and
hence the group index Ng depend on the wavelength of
light. Such materials are called dispersive.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Index
Refractive index n and the group index Ng of pure SiO2
(silica) glass are important parameters in optical fiber design
in optical communications. Both of these parameters depend
on the wavelength of light. Around 1300 nm, Ng is minimum
which means that for wavelengths close to 1300 nm, Ng is
wavelength independent. Thus, light waves with wavelengths
around 1300 nm travel with the same group velocity and do
not experience dispersion. This phenomenon is significant in
the propagation of light in glass fibers used in optical
communications.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive index n and the group index Ng of pure SiO2 (silica) glass as a
function of
wavelength.
Fig 9.6
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Example: Group velocity
Consider two sinusoidal waves which are close in
frequency, that is, waves of frequencies  and  +
. Their wavevectors will be kk and k + k. The
resultant wave will be
Ex(z,t) = Eocos[()t(kk)z] + Eocos[( + )t(k
+ k)z]
By using the trigonometric identity cosA + cosB =
2cos[1/2(AB)]cos[1/2(A + B)] we arrive at
Ex(z,t) = 2Eocos[()t(k)z]cos[tkz]
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
This represents a sinusoidal wave of frequency  which is
amplitude modulated by a very slowly varying sinusoidal of
frequency . The system of waves, that is, the modulation,
travels along z at a speed determined by the modulating term,
cos[()t(k)z]. The maximum in the field occurs when
[()t(k)z] = 2m = constant (m is an integer), which
travels with a velocity
dz 

dt k
or
d
vg 
dk
This is the group velocity of the waves since it determines the
speed of propagation of the maximum electric field along z.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Magnetic Field, Irradiance and Poynting Vector
The magnetic field (magnetic induction) component By
always accompanies Ex in an EM wave propagation.
If v is the phase velocity of an EM wave in an isotropic
dielectric medium and n is the refractive index, then
c
Ex  vBy  By
n
where v = (oro)1/2 and n = 
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
A plane EM wave traveling along k crosses an area A at right angles to the direction of
propagation. In time t, the energy in the cylindrical volume At (shown dashed) flow
through A.
Fig 9.7
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Energy Density in an EM Wave
As the EM wave propagates in the direction of the
wavevector k, there is an energy flow in this direction. The
wave brings with it electromagnetic energy.
The energy densities in the Ex and By fields are the same,
1
1 2
2
 o  r Ex 
By
2
2 o
The total energy density in the wave is therefore orEx2.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Poynting Vector and EM Power Flow
If S is the EM power flow per unit area,
S = Energy flow per unit time per unit area
( Avt )( o r E x2 )
S
 v o r E x2  v 2 o r E x B y
At
In an isotropic medium, the energy flow is in the direction
of wave propagation. If we use the vectors E and B to
represent the electric and magnetic fields in the EM wave,
then the EM power flow per unit area can be written as,
S = v2orEB
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Poynting Vector and Intensity
where S, called the Poynting vector, represents the energy
flow per unit time per unit area in a direction determined by
EB (direction of propagation). Its magnitude, power flow
per unit area, is called the irradiance (instantaneous
irradiance, or intensity).
The average irradiance is
I  Saverage  v o r E
1
2
2
o
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Average Intensity
Since v = c/n and r = n2 we can write
I  Saverage  c o nE  (1.3310 )nE
1
2
2
o
3
2
o
The instantaneous irradiance can only be measured if the
power meter can respond more quickly than the
oscillations of the electric field. Since this is in the optical
frequencies range, all practical measurements yield the
average irradiance because all detectors have a response
rate much slower than the frequency of the wave.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)