chapter 1 wave nature of light -...

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


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.

https://en.wikipedia.org/wiki/Physics

https://en.wikipedia.org/wiki/Locus_(mathematics)

https://en.wikipedia.org/wiki/Point_(geometry)

https://en.wikipedia.org/wiki/Phase_(waves)

https://en.wikipedia.org/wiki/Line_(mathematics)

https://en.wikipedia.org/wiki/Curve

https://en.wikipedia.org/wiki/Surface

https://en.wikipedia.org/wiki/Wave

https://en.wikipedia.org/wiki/Plane_wave

https://en.wikipedia.org/wiki/Plane_(mathematics)

https://en.wikipedia.org/wiki/Lens_(optics)

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


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


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.

https://www.rp-photonics.com/laser_beams.html

https://www.rp-photonics.com/beam_radius.html

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


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


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


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


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.



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 .



50

Snell’s Law

Figure 1.9


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).)


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.


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


(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


(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

chapter 1 wave nature of light -...

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