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Chapter 1 Wave Nature of Light
Prepared by Ömer Lütfi ÜNSAL and Dr. Beşire GÖNÜL
OPTOELECTRONICS AND PHOTONICS
Princibles and Practises
(S.O. KASAP)
CONTENT
1. Wave nature of light.
2. Dielectric waveguides.
3. Elements of solid state physics.
4. Semiconductor science.
5. P-N Junction.
6. Homojunction and heterojunction.
7. Light emitting diodes.
2
The wave nature of light is well recognized by interference and diffraction
phenomena.
Light an electromagnetic (EM) wave with time-varying electric and magnetic
fields, Ex and By .
Ex and By propagates through space in such a way that they are always perpendicular
to each other and the direction of propagation z.
3
1.1 LIGHT WAVES IN A HOMOGENEOUS MEDIUM
Plane electromagnetic Wave
4
An electromagnetic wave in a homogenous and isotropic medium is a traveling wave that has time-varying
electric and magnetic fields which are perpendicular to each other and the direction of propagation z. This is
a snapshot at a given time of a particular harmonic or a sinusoidal EM wave. At a time δt later, a point on
the wave, such as the maximum field, would have moved a distance vδt in the z-direction.
Figure 1.1
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
The simplest traveling wave is a sinusoidal wave that, for propagation along z, has the
general mathematical form
Ex electric field at position z at time t,
k wave vector (or propagation vector),
k propagation constant (the magnitude of k) and given by 2π /λ,
5
1.1 LIGHT WAVES IN A HOMOGENEOUS MEDIUM
Plane electromagnetic Wave
cos ( - )x o oE E t kz Eq. (1.1)
λ wavelength,
ω the angular frequency,
Eo amplitude of the wave, and
ϕo 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.
(ωt – kz + ϕo) is called the phase of the wave and denoted by ϕ.
Equation (1.1) describes a monochromatic plane wave of infinite extent traveling in
the positive z direction as depicted in Figure 1.2
6
cos ( - )x o oE E t kz Eq. (1.1)
7
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.
Figure 1.2
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
8
In physics, a wavefront is the locus of points characterized by
propagation of position of the same phase: a propagation
of a line in 1d, a curve in 2d or a surface for a wave in 3d.
The wavefronts of a plane wave are
planes. A lens can be used to
change the shape of
wavefronts. Here, plane
wavefronts become
spherical after going
through the lens.
A surface over which the phase of a wave is constant at
a given instant is referred to as a wavefront.
Time-varying magnetic fields result in time-varying
electric fields (Faraday’s law) and vice versa.
The optical field refers to the electric field Ex.
9
During a time interval δt, constant phase (and hence the maximum field) moves a
distance δz.
The phase velocity of this wave is therefore δz/δt.
Thus the phase velocity v is
10
z
t k
v Eq. (1.4)
υ is the frequency (ω= 2πυ) of the EM wave.
For an EM wave propagating in free space v is the speed of light in vacuum or c.
When propagation vectors are all parallel and the plane wave propagates without the
wave diverging; the plane wave has no divergence.
EM waves must obey a special wave equation that describes the time and space
dependence of the electric field. In an isotropic and linear dielectric medium, the
relative permittivity (εr) is the same in all directions and is independent of the electric
field. The field E in such a medium obeys Maxwell’s EM wave equation
11
2 2 2 2
2 2 2 20o r o
E E E E
x y z t
Eq. (1.5)
in which µo is the absolute permeability, εo is the absolute permittivity, and εr is the
relative permittivity of the medium.
Maxwell’s Wave Equation and Diverging Waves
12
Examples of possible EM waves. (a) A perfect plane wave. (b) A perfect spherical wave. (c) A divergent
beam.
Figure 1.4
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
A spherical wave is described by a traveling field that emerges from a point EM
source and whose amplitude decays with distance r from the source. At any point r
from the source, the field is given by
13
cos ( - )A
E t krr
Eq. (1.6)
in which A is constant.
Many light beams, such as the output from a laser, can be described by assuming that
they are Gaussian beams.
14
(a) Wavefronts of a Gaussian light beam. (b) Light intensity across a beam cross-section. (c) Light intensity
vs. radial distance r from beam axis (z).
Figure 1.5
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
15
The beam waist (or beam focus) of a
Gaussian beam is the location along the
propagation direction where the beam
radius has a minimum.
The waist radius is the beam radius at this
location.
The amplitude of the beam varies spatially away from the beam axis
and also along the beam axis.
The beam diameter 2w at any point z is defined in such a way that the
cross-sectional area πw2 at that point contains 86% of the beam
power.
The beam diameter 2w increases as the beam travels along z.
16
17 Example 1.1
Consider a He-Ne laser beam at 633 nm with a spot size of 1 mm.
Assuming a Gaussian beam, what is the divergence of the beam?
A Diverging Laser Beam
Solution of Example 1.1
9
4
3
4 633 1042 8.06 10 0.046
2 1 10
o
o
mrad
w m
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 relative permittivity measures the ease with which the medium becomes
polarized and hence it indicates the extent of interaction between the field and the
induced dipoles.
In a dielectric medium of relative permittivity εr, the phase velocity υ is given by
18
1
o r o v Eq. (1.7)
εr depends on the frequency.
1.2 REFRACTIVE INDEX AND
DISPERSION
Typical frequencies that are involved in optoelectronic devices are in the infrared
(including far infrared), visible, and UV,
Optical frequencies; roughly 1012 Hz to 1016 Hz.
For EM wave traveling in free space, εr= 1 and vvacuum = 1/(εoµo)1/2 = c = 3× 108 m s-1,
the velocity of light in vacuum.
The ratio of speed of light in free space to its speed in a medium is called the
refractive index n of the medium, that is,
19
r
cn
vEq. (1.8)
Light propagates more slowly in a denser medium that has
a higher refractive index.
20
r
cn
v
If the k is the propagation constant (k = 2π / λ) and λ is the wavelength, both in free
space, then in the medium
k medium = nk and
λ medium= λ /n.
The refractive index of a medium is not necessarily the same in all directions.
21
In noncrystalline 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.
In crystals, however, the atomic arrangements and interatomic bonding are different
along different directions.
Crystals, in general, have nonisotropic, or anisotropic, properties.
22
In general, the refractive index n seen by a propagating electromagnetic wave in a
crystal will depend on the value of εr along the direction of the oscillating electric
field (i.e., along the direction of polarization).
Depending on the crystal structure, the relative permittivity εr is different along
different crystal directions.
For example, suppose that the wave in Figure 1.1 is traveling along the z- direction in
a particular crystal with its electric field oscillating along the x-direction.
If the relative permittivity along this x-direction is εrx , then nx = (εrx)1/2. The wave
therefore propagates with a phase velocity that is c / nx .
23
The variation of n with direction of propagation and the direction of the
electric field depends on the particular crystal structure.
With the exception of cubic crystals (such as diamond), all crystals
exhibit a degree of optical anisotropy that leads to a number of important
applications.
Typically, noncrystalline solids, such as glasses and liquids, and cubic
crystals are optically isotropic; they possess only one refractive index
for all directions.
24
Relative permittivity εr or the dielectric constant of materials, in general, depends
on the frequency of the electromagnetic wave.
The relationship n = (εr)1/2 between the refractive index n and εr must be applied at
the same frequency for both n and εr.
The relative permittivity for many materials can be vastly different at high and low
frequencies because different polarization mechanisms operate at these
frequencies.
25Relative permittivity εr
At low frequencies (e.g. 60 Hz or 1kHz) all polarization mechanisms
present can contribute to εr , whereas at optical frequencies only the
electronic polarization can respond to the oscillating field.
Electronic polarization involves the displacement of light electrons
wrt heavy positive ions in the crystal.
26
The relative permittivity depends on the polarizability α per molecule (or atom) in
the solid.
α is defined as the induced electric dipole moment per unit applied field.
The expression for the relative permittivity is
27
1r
o
N
Eq. (1.9)
in which N is the number of molecules per unit volume.
Both the atomic concentration, or density, and polarizability therefore increase n.
For example, glasses of given type but with greater density tend to have higher n.
What factors affect n ?
Since 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 z-direction.
Figure 1.6 shows how two perfectly harmonic waves of slight different
frequencies ω – δω and ω + δω interfere to generate a periodic wave
packet that contains an oscillating field at the mean frequency ω that is
amplitude modulated by a slowly varying field of frequency δω.
28
1.2 GROUP VELOCITY AND GROUP INDEX
29
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.
Figure 1.6
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
We are interested in the velocity of this wave packet. The two sinusoidal waves of
frequencies ω – δω and ω + δω will propagate with propagation constants k – δk and
k + δk respectively inside the material so that their sum will be
30
, cos cosx o oE z t E t k k z E t k k z
Eq. (1.10)
By using the trigonometric identity
we arrive at
1 1
cos cos 2cos cos2 2
A B A B A B
, 2 cos cosx oE z t E t k z t kz Eq. (1.11)
This represents a sinusoidal wave of frequency ω, which is amplitude
modulated by a very slowly varying sinusoid of frequency δω.
The system of waves, that is, the modulation, travels along z at a speed
determined by the modulating term, cos[(δω)t – (δk)z].
The maximum in the field occurs when [(δω)t – (δk)z] = 2mπ =
constant (m is an integer), which travels with a velocity
31
z
t k
or g
k
v
Eq. (1.12)
vg determines the speed of propagation of the maximum electric field
along z.
vg represents the speed with which energy or information is
propagated since it defines the speed of the envelope of the amplitude
variation.
the velocity vg is the group velocity of the waves,
The maximum electric field in Figure 1.6 advances with a velocity vg
whereas the phase variations in the electric field propagate at the phase
velocity v.
32
ω = vk and phase velocity is v =c/n
In vacuum v is simply c and independent of wavelength.
Thus for waves travelling in vacuum, ω = ck and the group velocity is
33
vacuum phase velocityg ck
v
Eq. (1.13)
In vacuum or air, the group velocity is the same as the phase velocity.
For an EM wave in a medium, k in Eq. (1.13) is the propagation constant inside the
medium, which can be written k = 2πn /λo is the free pace wavelength.
The group velocity then is not necessarily the same as the phase velocity v, which
depends on ω/k and is given by c/n.
The group velocity vg, on the other hand, is δω/δk, which depends on how the
propagation changes in the medium, δk, with the change in frequency δω, and δω/δk,
is not necessarily the same as ω/k when the refractive index has a wavelength
dependence.
34
Suppose that the refractive index n=n( λo) is a function of (free space)
wavelength λo.
Its gradient would be dn/d λo.
We can easily find the group velocity, as shown in Example 1.2, by first
finding δω and δk in terms of δn and d λo, and then using Eq. (1.13),
35
mediumg
o
o
c
k dnn
d
v
Eq. (1.14)
This can be written as
36
mediumg
g
c
Nv Eq. (1.15)
in which
g o
o
dnN n
d
Eq. (1.16)
is defined as the group index of the medium.
Equation (1.16) defines the group refractive index Ng of a medium and
It determines the effect of the medium on the group velocity.
What is important in Eqs. (1.15) and (1.16) is the gradient of the refractive index,
dn/dλo.
If the refractive index is constant and independent of the wavelength, at least
over the wavelength range of interest, then Ng = n; and the group and phase
velocities are the same.
In general, for many materials the refractive index n and hence the group index Ng
depend on the wavelength of light by virtue of εr being frequency dependent.
Then, both the phase velocity v and the group velocity vg depend on the wavelength
and the medium is called a dispersive medium.
37
The 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 as shown in
Figure 1.7.
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.
38
39
Refractive index n and the group index Ng of pure SiO2 (silica) glass as a function of
wavelength.
Figure 1.7
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
40 Example 1.2
Consider two sinusoidal waves that are close in frequency, that is, waves of
frequencies ω – δω and ω + δω as in Figure 1.4. Their propagation constant will be
k – δk and k + δk . The resultant wave will be
Group Velocity
, cos cosx o oE z t E t k k z E t k k z
By using the trigonometric identity
we arrive at
, 2 cos cosx oE z t E t k z t kz
1 1
cos cos 2cos cos2 2
A B A B A B
41 Example 1.2
As illustrated in Figure 1.6, this represents a sinusoidal wave of frequency ω,
which is amplitude modulated by a very slowly varying sinusoid of frequency
δω.
The system of waves, that is, the modulation, travels along z at a speed determined
by the modulating term, cos[(δω)t – (δk)z].
The maximum in the field occurs when [(δω)t – (δk)z] = 2mπ = constant (m is an
integer), which travels with a velocity
Group Velocity
or g
z
t k k
v
This is the group velocity of the waves, as stated in Eq. (1.11), since it determines
the speed of propagation of the maximum electric field along z.
42 Example 1.3
Consider ω= 2πc / λo and k = 2πn / λo ,
where λo is the free-space wavelength.
By finding expressions for dω and dk in terms of dn
and dλo derive Eq. (1.15) for the group velocity vg.
Group Velocity and Index
43 Solution of Example 1.3
Differentiate ω= 2πc /λo to get dω = –(2πc/λo2)dλo, and then differentiate k = 2πn /λo to
find
2 2 2 1 / 2 / 2 /o o o o o o o
o o
dn dndk n d d n d
d d
We can now substitute for dω and dk in Eq. (1.12),
2
2
2 /
2 /
o o
oo o ooo
g
c d c
dndn nnd
kd
d
v
44 Example 1.4
Consider a light wave traveling in a pure SiO2 (silica) glass
medium. If the wavelength of light is 1 µm and the refractive
index at this wavelength is 1.450, what is the phase velocity,
group index (Ng), and group velocity (vg)?
Group and Phase
Velocities
45 Solution of Example 1.4
The phase velocity is given by
8 1 8 1 3 10 m s 1.450 2.069/ 10 m sc n v
The group velocity is about ∼0.9%, smaller than the phase velocity.
8 1 8 1 / 3 10 m s 1.463 2.051 10 m sgg c N v
From Figure 1.7, at λ = 1 µm , Ng = 1.463, so that
46
A vibrating object is wiggling about a fixed position. Like the mass on a spring in the animation
Amplitude of Vibration
The final measurable quantity that describes a
vibrating object is the amplitude. The amplitude is
defined as the maximum displacement of an object
from its resting position. The resting position is that
position assumed by the object when not vibrating.
Once vibrating, the object oscillates about this fixed
position. If the object is a mass on a spring (such as
the discussion earlier on this page), then it might be
displaced a maximum distance of 35 cm below the
resting position and 35 cm above the resting position.
In this case, the amplitude of motion is 35 cm.
47
A light wave traveling in a medium with a greater refractive index (n1 > n2) suffers reflection and refraction
at the boundary. (Notice that λt t is slightly longer than λ.)
Figure 1.8
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
n2
1.4 SNELL’S LAW AND TOTAL
INTERNAL REFLECTION (TIR)
Consider a traveling plane EM wave in a medium
(1) of refractive index n1 propagating toward a
medium (2) with a refractive index n2.
When the wave reaches the plane boundary
between the two media,
i. a transmitted wave in medium 2 and
ii. a reflected wave in medium 1 appear.
The transmitted wave is called the refracted light.
481.4 SNELL’S LAW AND TOTAL
INTERNAL REFLECTION (TIR)
The angles;
Ɵi defines direction of the incident waves,
Ɵt defines direction of the transmitted waves,
Ɵr defines direction of the reflected waves,
respectively, with respect to the normal to the
boundary plane.
Since both the incident and reflected waves
are in the same medium, the magnitudes of kr
and ki are the same, kr = ki .
491.4 SNELL’S LAW AND TOTAL
INTERNAL REFLECTION (TIR)
50
Snell’s Law
Figure 1.9
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
2
1
i
t
Sin n
Sin n
Simple arguments based on constructive
interference can be used to show that there can
be only one reflected wave that occurs at an
angle equal to the incidence angle.
The two waves along Ai and Bi are in phase.
When these waves are reflected to become
waves Ar and Br then they must still be in
phase, otherwise they will interfere
destructively and destroy each other.
51
if Ɵi = Ɵr these two
waves can stay in phase.
All other angles lead to the
waves Ar and Br being out of
phase and interfering
destructively.
52
The refracted waves propagates in
medium 2 with a refractive index n2 .
Hence refractive waves have different
velocities than that of incident ones.
What happens to a wavefront AB as it
propagates from medium 1 to 2?
(The points A and B on this front are always
in phase)
53
The wavefront AB thus becomes
the front A'B' in medium 2.
Unless the two waves at A' and B'
still have the same phase, there
will be no transmitted wave.
A' and B' points on the front are in
phase only for one particular
transmitted angle, Ɵt .
54
If it takes time t for the phase at B on wave
Bi to reach B', then BB' = v1t = ct / n1.
During this time t, the phase A has
progressed to A' where AA' = v2t = ct / n2.
A' and B' belong to the same front just like
A and B so that AB is perpendicular to ki
in medium 1 and A'B' is perpendicular
to kt in medium 2.
55
From geometrical considerations,
AB' = BB' / sinƟi and AB' = AA' / sinƟt
so that
56
1 2 1 2
2 1
sin' or
sin sin sin
i
i t t
t t nAB
n
v v v
v
Eq. (1.17)
This is Snell’s law, which relates the
angles of incidence and refraction to the
refractive indices of the media.
If we consider the reflected wave, the wavefront
AB becomes A''B' in the reflected wave.
In time t, phase B moves to B' and A moves to
A''.
Since they must still be in phase to constitute
the reflected wave, BB' must be equal to AA''.
57
Suppose it takes time t for the wavefront B
to move to B' (or A to A'').
Then, since BB' = AA'' = v1t , from
geometrical considerations,
58
1 1'sin sini r
t tAB
v v
Eq. (1.18)
So that Ɵi = Ɵr .
Angles of incidence and reflection are the same.
When n1 > n2 then obviously the transmitted angle is greater than the incidence
angle.
When the refraction angle Ɵt reaches 90°, the incidence angle is called the critical
angle Ɵc ,
59
Eq. (1.19)
When the incidence angle Ɵi exceeds Ɵc
there is no transmitted wave but only a reflected wave.
This phenomenon is called as total internal reflection (TIR).
2
1
sin c
n
n
60
Light wave travelling in a more dense medium strikes a less dense medium.
Depending on the incidence angle with respect to Ɵc, which is determined by the ratio of the refractive
indices, the wave may be transmitted (refracted) or reflected.
(a) Ɵi < Ɵc (b) Ɵi = Ɵc (c) Ɵi > Ɵc and total internal reflection.
(Wavefronts are only indicated in (a).)
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
Figure 1.10
61
Light wave traveling in a more dense medium strikes a less dense medium. The plane of incidence is the
plane of the paper and is perpendicular to the flat interface between the two media. The electric field is
normal to the direction of propagation. It can be resolved into perpendicular and parallel components.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA
Figure 1.11
As apparent from Figure 1.11, the incident, transmitted, and reflected wave all
have a wave vector component along the z-direction; that is, they have an effective
velocity along z.
The fields 𝑬𝒊,⊥, 𝑬𝒓,⊥and 𝑬𝒕,⊥are all perpendicular to the z-direction.
These waves are called transverse electric field (TE) waves.
Thewaves with 𝑬𝒊,⫽, 𝑬𝒓,⫽and 𝑬𝒕,⫽have only their magnetic field components
perpendicular to the z-direction, and these are called transversemagnetic field
(TM) waves.
62
1.5 FRESNEL’S EQUATIONS
Amplitude Reflection and Transmission Coefficients (r And t)
We will describe the incident, reflected, and refracted waves each by the exponential
representation of a traveling wave:
63
expi ioE E j t k ri Eq. (1.20)
expr roE E j t k rr Eq. (1.21)
expt toE E j t k rt Eq. (1.22)
in which r is the position vector, the wave vectors ki , kr , and kt describe the
directions of the incident, reflected, and transmitted waves, and Eio , Ero , and Eto are
the respective amplitudes.
Any phase changes such as ϕr and ϕt in the reflected and transmitted waves with
respect to the phase of the incident wave are incorporated into the complex
amplitudes, Ero and Eto . The reflected wave is then said to be linearly polarized
because it contains electric field oscillations that are contained within a well-
defined plane, which is perpendicular to the plane of incidence and also to the
direction of propagation.
Electric field oscillations in unpolarized light, on the other hand, can be in any one
of infinite number of directions that are perpendicular to the direction of
propagation.
64
In linearly polarized light, however, the field oscillations are contained within a
well-defined plane.
Light emitted from many light sources, such as a tungsten light bulb or an LED
diode, is unpolarized.
Unpolarized light can be viewed as a stream or collection of EM waves whose
fields are randomly oriented in a direction that is perpendicular to the direction of
light propagation.
65
To calculate the intensity or irradiance of the reflected and transmitted waves when
light traveling in a medium of index n1 is incident at a boundary where the refractive
index changes to n2.
For a light wave traveling with a velocity v in a medium with relative permittivity εr, the
light intensity I is defined in terms of the electric field amplitude Eo as
66 Intensity, Reflectance, And Transmittance
21
2r o oI E v Eq. (1.23)
67 Intensity, Reflectance, And Transmittance
21
2r o oI E v
Here ½ εrεoEo2 represents the energy in the field per unit volume.
When multiplied by the velocity v it gives the rate at which energy is transferred
through a unit area.
Since v = c/n and εr = n2 , the intensity is proportional to nEo2.
Reflectance R measures the intensity of the reflected light with respect to that of
the incident light.
68
2
1 2//
1 2
n n
n n
R R R Eq. (1.24)
Since a glass medium has a refractive index of around 1.5 this means that
typically 4% of the incident radiation on an air–glass surface will be reflected
back.
For normal incidence, the incident and transmitted beams are normal and the
transmittances
69
1 2
// 2
1 2
4n n
n n
T T T Eq. (1.25)
Further, the fraction of light reflected and fraction transmitted must add to unity.
Thus R + T = 1.
Transmittance T relates the intensity of the transmitted wave to that of the
incident wave.
70 Example 1.5
Consider the reflection of light at normal incidence on a boundary between a glass
medium of refractive index 1.5 and air of refractive index 1.
(a) If light is traveling from air to glass, what is the reflection coefficient and the
intensity of the reflected light?
(b) If light is traveling from glass to air, what is the reflection coefficient and the
intensity of the reflected light?
Reflection at Normal Incidence, and Internal
and External Reflection
71 Solution of Example 1.5
(a)The light travels in air and becomes partially reflected at the surface of the glass that
corresponds to external reflection. Thus n1 = 1 and n2 = 1.5. Then
// 0.0.4 or 4% R r
This is negative, which means that there is a 180° phase shift.
The reflectance (R), which gives the fractional reflected power, is
1 2//
1 2
1 1.50.2
1 1.5
n n
n n
r r
2
1 2//
1 2
n n
n n
R R R
72 Solution of Example 1.5
(b) The light travels in glass and becomes partially reflected at the glass–air interface
that corresponds to internal reflection. Thus n1 = 1.5 and n2 = 1. Then
There is no phase shift. The reflectance is again 0.04 or 4%.
In both cases (a) and (b), the amount of reflected light is the same.
1 2//
1 2
1.5 10.2
1.5 1
n n
n n
r r
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