UNIT 1 - INTRODUCTION By: Ajay Kumar Gautam

Asst. Prof.

Electronics & Communication Engineering

Dev Bhoomi Institute of Technology & Engineering, Dehradun

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Syllabus

• Demand of Information Age, Block Diagram of Optical fiber Communication System, Technology used in OFC System, Structure and types of Fiber, modes and Configuration,

• mode theory for circular guide modal equation, modes in optical fiber, linearly polarized modes, attenuation factors, pulse broadening in optical fiber,

• single mode fiber, mode field diameter, signal distortion in single mode fiber, Derivation of material dispersion and waveguide dispersion. Attenuation, Signal Degradation in Optical Waveguides, Pulse Broadening in Graded index fiber Waveguides, Mode Coupling

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Communication System

• Transfer of information from one place to another.

• A communication system is usually required to convey

information over any distance.

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General Communication

System 29 July 2012 UNIT 1

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• Its function is to convey the signal from information source

over the transmission medium to the destination.

• The communication system consists of

1) Transmitter or modulator linked to information source.

2) Transmission medium

3) Receiver or demodulator at the destination end.

• In electrical communication systems the information source

provides an electrical signal.

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• This electrical signal is usually derived from a message

signal which is not electrical ( e.g. sound.).

• The electrical/electronic components, then converts the

signal into a suitable form for propagation over the

transmission medium.

• This is achieved by modulating a carrier.

• Carrier may be an electromagnetic wave.

• The transmission medium can consist of a pair of wires, a

coaxial cable or a radio link through free space.

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• However in any transmission medium the signal is

attenuated or suffer loss.

• So, in any communication system there should be a

maximum permitted distance between the transmitter and

the receiver.

• So, in order to reduce the losses , remove the signal

distortion & to increase the signal level before transmission

is continued down, the link is required to install the

repeaters or line amplifiers.

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Optical Communication

System 29 July 2012 UNIT 1

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In OFC system,

• The information source provides an electrical signal to the transmitter.

• The optical source provides modulation of the lightwave carrier.

• The optical source may be semiconductor Laser or LED.

• The transmission medium consists of the optical fiber cable.

• The receiver consists of an optical detector.

• The receiver derives a further electrical stage, means demodulation of the optical carrier.

• The optical detectors may be photodiodes, phototransistors or photoconductors.

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Advantages of OFC

System

Following advantages of optical fiber over copper wires.

a) Long distance transmission

b) Large information capacity

c) Small size and low weight

d) Immune to electrical interference

e) Enhanced safety

f) Increased signal security

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a. Long Distance

Transmission

• Since the optical fibers have lower transmission losses

compared to copper wires, so the data can be sent over long

distances, so optical fiber reduces the no. of intermediate

repeaters.

• The repeaters are not required to boost up and to restore

the signals in long transmission.

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b. Large information

Capacity

• Optical fibers have wider bandwidths than copper wires, so

that more information can be sent over a single physical

line.

• This is useful to decrease the number of physical lines

needed for sending a given amount of information.

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c. Small Size and Low

Weight

• Since the optical fiber has low weight and small dimensions,

so it is useful in aircraft, satellite and ships, where small and

light weight cables are advantageous.

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d. Immune to Electrical

Interference

• Since the optical fiber is dielectric material, i.e. it does not

conduct electricity.

• So, the optical fibers are immune to the electromagnetic

interference effects.

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e. Enhanced Safety

• Optical fibers offers high degree of operational safety,

because they does not have problems of ground loops,

sparks and high voltages.

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f. Increased Signal Security

• Since the optical signal is well confined within the fiber and

we have an opaque coating around the fiber, which is useful

to absorb any signal emissions.

• It means optical fiber offers a high degree of data security.

• So, the optical fibers are useful in applications where

information security is important.

• Ex: financial, government, legal and military systems.

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Refractive index

• It is defined as the ratio of the velocity of light in a vacuum

to the velocity of light in that medium. i.e.,

n=c/v

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Reflection & Refraction

• Let we are having two different attached media.

• If the light ray enters the boundary that separates the two

different media.

• Then, there are two rays. One part will be reflected back

into the first medium. Other refracted into second medium.

• The Snell’s law gives a relation at the interface of these two

medium, and is given by: n1sinφ1=n2sin φ2.

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Cond…

•

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Fig: Refraction and Reflection of a light ray at material

boundary

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Modes and

Configuration

1. Fiber types

2. Rays and modes

3. Step index fiber

4. Ray optics representation

5. Wave representation in a dielectric slab waveguide

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1. Fiber types

• The optical fiber a dielectric waveguide, that operates at

optical frequencies.

• Optical fiber waveguide is normally cylindrical in form.

• Since, the light is the electromagnetic wave, so the fiber

waveguide confines electromagnetic energy in the form of

light.

• The direction of propagation of light is parallel to the fiber

axis & is represented by z-direction.

• A set of guided electromagnetic wave called the modes of

the waveguide.

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Cond…

• The modes are used to describe the propagation of the

light along the waveguide.

• These guided modes are referred to as the bounded or trapped

modes of the waveguide.

• Each guided mode is a pattern of electric & magnetic field

distributions.

• This pattern is usually repeated along the fiber at equal

intervals.

• The most widely used optical fiber waveguide is shown on

next slide.

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Fig: conventional silica fiber structure.

ncore > nCladding.

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Cond…

• It consists of single solid dielectric cylinder of radius a, and

refractive index n1.

• This is called core of the fiber waveguide.

• The core is surround by a solid dielectric region called

cladding with refractive index n2.

• The purpose of cladding is not to propagate the light.

• It has some other purposes.

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Cond…

• The cladding is useful because of the following points.

1. It reduces the scattering losses.

2. It adds mechanical strength to the fiber.

• It protects the core from absorbing surface contaminations.

• The core is made up of highly pure silica glass (SiO2).

• There are basically two kind of optical fibers. Step index &

graded index fiber.

• In step index fiber, the refractive index of the core is uniform

& undergoes an abrupt change (or step) at cladding

boundary.

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Cond…

• If the refractive index of the core varies as a function of

the radial distance from the center of the fiber, this is called

graded-index fiber.

• Step-index and graded-index fibers can be further classified

into single-mode and multi-mode classes.

• Single-mode fiber contains one mode of propagation &

multi-mode fiber contains many hundred of modes.

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Fig: single mode and multi-mode step-index and

graded-index optical fiber

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Cond…

• Advantage of Multimode Fiber:

1. It is easy to launch optical power into multimode fiber.

2. LED can be used to launch the optical power.

• Disadvantages of Multimode Fiber:

1. It suffers intermodal dispersion

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2. Rays and Modes

• The guided modes consists of a set of simple

electromagnetic field configurations.

• The mode travelling in the positive z direction is given by:

ej(ωt-βz)

• β is the z component of the wave propagation constant

k = 2π/λ

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3. Step-Index fiber

Structure

• In practical step-index fiber, the refractive index of the clad

region is given by: n2 = n1(1-∆).

• Usually ∆=0.01.

• ∆ is called core-cladding index difference.

• The electromagnetic energy propagates along the optical

waveguide through total internal reflection.

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4. Ray Optics

Representation

• We know that, core size of the multimode fiber is much

larger than the wavelength of the light.

• The wavelength of the light is usually 1µm.

• Consider multimode step index fiber with core radius 25 to

100µm.

• An ideal multimode step index optical waveguide can be

seen by a simple ray optics representation.

• There are two type of rays that propagate into this kind of

fiber. 1. Meridional rays 2. skew rays.

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Cond…

• Meridional rays are confined to the meridian planes of the

fiber.

• Meridian plane is that plane of the fiber, which contains

axis of the symmetry of the fiber (core axis).

• Skew rays are not confined to a single plane, they follow

helical path along the fiber.

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Fig: Ray optics Representation of Skew Rays

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Fig: Meridional Rays Optics Representation

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Cond…

• Meridional ray is shown on previous slide for a step-index

fiber.

• The light ray enters the fiber axis and strikes the core-

cladding interface at a normal angle φ. If the total internal

reflection occurs in this case, then the meridional ray follow

a zigzag path along the fiber core.

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Cond…

•

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Mode Theory for

Circular Waveguides

• A mode, in this sense, is a spatial distribution of optical

energy in one or more dimensions that remains constant in

time.

• The mode theory, along with the ray theory, is used to

describe the propagation of light along an optical fiber.

• A set of guided electromagnetic waves is called

the modes of the fiber.

• If the mode is no longer bound to the fiber, it is said to be

cut off mode.

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Cond…

• However, an optical fiber is always able to propagate at least

one mode.

• This mode is referred to as the fundamental mode of the

fiber.

• The fundamental mode can never be cut off.

• An optical fiber that operates above the cutoff wavelength

is called a single mode fiber.

• An optical fiber that operates below the cutoff wavelength

is called a multimode fiber.

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Cond....

• Maxwell's equations describe electromagnetic waves or

modes as having two components.

• The two components are the electric field, E (x, y, z), and

the magnetic field, H(x, y, z).

• The electric field E and the magnetic field H are at right

angles to each other.

• Modes traveling in an optical fiber are said to be transverse.

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Cond…

• The transverse modes, shown in Fig., propagate along the

axis of the fiber.

• The mode field patterns shown in Fig. are said to be

transverse electric (TE).

• In TE modes, the electric field is perpendicular to the

direction of propagation.

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Fig. Electric field distribution for several of lower - order guided modes

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Cond…

• The magnetic field is in the direction of propagation.

• Another type of transverse mode is the transverse magnetic

(TM) mode.

• TM modes are opposite to TE modes.

• In TM modes, the magnetic field is perpendicular to the

direction of propagation. The TE mode field patterns

shown in Fig. indicate the order of each mode.

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Cond…

• The order of each mode is indicated by the number of field

maxima within the core of the fiber.

• For example, TE0 has one field maxima.

• The electric field is a maximum at the center of the

waveguide and decays toward the core-cladding boundary.

• TE0 is considered the fundamental mode or the lowest

order standing wave.

• As the number of field maxima increases, the order of the

mode is higher.

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Maxwell’s Equations

• To analyze the optical waveguide we need to consider

Maxwell’s equations that give the relationships between the

electric and magnetic fields.

• Assuming a linear dielectric material having no currents and

free charges, these equations take the form,

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Cond… 29 July 2012 UNIT 1

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(1 )

(1 )

. 0 (1 )

. 0 (1 )

BX E a

t

DX H b

t

D c

B d

Cond…

• Where D = E and B = µH. The parameter is the

permittivity (or dielectric constant) and µ is the permeability

of the medium.

• Taking the curl of Eq. (1a) and making use of Eq. (1b)

gives,

• Using the vector identity

• And using Eq. (1c), Eq. (2a) becomes

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2

2( ) ( ) (2 )

EX X E X H a

t t

2( ) ( . )X X E E E

Cond…

• Similarly, by taking the curl of Eq. (1b), it can be shown

that

• Equations (2b) and (2c) are the standard wave equations.

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22

2(2 )

EE b

t

22

2(2 )

HH c

t

Waveguide Equations

• If the electromagnetic waves are to propagate along the z

axis, they will have a functional dependence of the form

• Which are harmonic in time t and coordinate z.

• The parameter β is the z component of the propagation

vector and will be determined by the boundary conditions

on the electromagnetic fields at the core – cladding

interface

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( )

0

( )

0

( , ) (3 )

( , ) (3 )

j t z

j t z

E E r e a

H H r e b

Cond…

• When Eq. (3a) and (3b) are substituted into Maxwell’s curl

equations, we have

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1(4 )

(4 )

1(4 )

zr

zr

rz

Ejr E j H a

r

Ej E j H b

r

ErE j H c

r r

Cond…

• And, from Eq. (1b),

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1(5 )

(5 )

1(5 )

zr

zr

rz

Hjr H j E a

r

Hj H j E b

r

HrH j E c

r r

Fig. 2 Cylindrical coordinate system used for analyzing electromagnetic wave

propagation in an optical fiber

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• when Ez and Hz are known, the remaining transverse components Er,

Eϕ, Hr and Hϕ can be determined.

• Hϕ or Er can be found in terms of Ez or Hz by eliminating Eϕ or Hr

from Eq. (4a) and Eq. (5b). These yields,

2

2

2

2

(6 )

(6 )

(6 )

(6 )

z zr

z z

z zr

z z

E HjE a

q r r

E HjE b

q r r

H EjH c

q r r

H EjH d

q r r

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Where q2 = 2 - β2 = k2 - β2.

• Substitution of Eq. (6c) and Eq. (6d) into Eq. (5c) gives the wave

equation in cylindrical coordinates [14 - 16],

• And substitution of Eq. (6a) and Eq. (6b) into Eq. (4c) gives,

2 2 22

2 2 2 2

1 10 (7 )z z z z

z

E E E Eq E a

r r r r z

2 2 22

2 2 2 2

1 10 (7 )z z z z

z

H H H Hq H b

r r r r z

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• Mode solutions can be obtained in which either Ez = 0 or Hz = 0.

• When Ez = 0 the modes are called transverse electric or TE modes, and

when Hz = 0 they are called transverse magnetic or TM modes.

• Hybrid mode exists if both Hz and Ez are nonzero. These are

designated as HE or EH a mode, depending on whether Hz and Ez,

respectively, can makes a larger contribution to the transverse field.

• The fact that the hybrid modes are present in optical waveguides makes

their analysis more complex than in simpler case of hollow metallic

waveguides where only TE and TM modes are found.

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UNIT 1

Wave Equations for Step –

Index Fibers

• We can use Eq. (7a) and Eq. (7b) to find the guided modes in a step –

index fiber.

• A standard mathematical procedure for solving equations such as Eq.

(7a) is to use the separation – of – variables method, which assumes a

solution of the form

• Assume the time- and z- dependent factors are given by

z 1 2 3 4r z t (8 )E AF F F F a

( )

3 4z t e (8 )j t zF F b

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UNIT 1

• We assume a periodic function of the form

• Thus constant can be positive or negative, but it must be an integer

since the field must be periodic in ϕ with a period of 2π.

• Now substituting Eq. (8c) and Eq. (8b) into Eq. (8a), the wave equation

for Ez [Eq. (7a)] becomes

• This is well - known differential equation for Bessel functions.

2( ) (8 )jF e c

2 221 1

12 2

10 (8 )

F Fq F d

r r r r

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UNIT 1

• Three different types of Bessel functions

If q is real then the solutions are

Bessel functions of first kind

Bessel functions of second kind

If q is imaginary, then the solutions are

Modified Bessel functions of first kind

Modified Bessel functions of second kind

If q is complex, then the solutions are

Hankel functions of first kind

Hankel functions of second kind

• The quantity is called the order of the function and (qr) is called the

argument of the Bessel function.

J qr

( )Y qr

/I qr j

( / )K qr j

(1)H qr

(2)H qr

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UNIT 1

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

qr

Jv(q

r)

J0(qr)

J1(qr)

J2(qr)

Fig. 3 Bessel functions of first kind - Matlab Result

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UNIT 1

0 1 2 3 4 5 6 7 8 9 10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

qr

Yv(q

r)

Y0(qr)

Y1(qr)

Y2(qr)

Fig. 4 Bessel functions of second kind - Matlab Result 29 July 2012

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UNIT 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

qr/j

I v(q

r/j)

I0(qr)

I1(qr)

I2(qr)

Fig. 5 Modified Bessel functions of first kind - Matlab Result

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UNIT 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

qr/j

Kv(q

r/j)

K0(qr)

K1(qr)

K2(qr)

Fig. 6 Modified Bessel functions of second kind - Matlab Result

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UNIT 1

• Bessel function of first kind: The functions are finite for all values of

r.

• Bessel function of second kind: The functions start from - ∞ at r =

0 and have finite values for all the other values of r.

• Modified Bessel function of first kind: The functions increases

monotonically with increases of r.

• Modified Bessel function of second kind: The functions decreases

monotonically with increases of r.

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UNIT 1

Therefore we can conclude that

• Bessel function of 1st kind is the appropriate solution for the

modal fields inside the core of an optical fiber.

where, with

• Modified Bessel function of 2nd kind is the appropriate

solution for the modal fields outside the core of an optical fiber.

where, with

uJ r

( )K wr

2 2 2

1u k

2 2 2

2w k

11

2 nk

22

2 nk

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UNIT 1

• The expressions for Ez and Hz inside the core are, when (r < a)

• The expressions for Ez and Hz outside the core are, when (r > a)

• where A, B, C & D are arbitrary constants which are to be evaluated

from the boundary conditions.

2

2

(9 )

(9 )

j j z j t

z

j j z j t

z

E CK wa e c

H DK wa e d

1

1

( ) (9 )

(9 )

j j z j t

z

j j z j t

z

E AJ ua e a

H BJ ua e b

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UNIT 1

• For a guided mode, the propagation constant lies between two limits k1

and k2.

• If then a field distribution is generated

which will has an oscillatory behavior in the core and a decaying

behavior in the cladding. The energy then is propagated along fiber

without any loss.

• Where is free – space propagation constant.

2 2 1 1n k k k n k

2k

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UNIT 1

Boundary Conditions

• The solutions for β must be determined from the boundary

conditions.

• The boundary conditions requires that the tangential

components Eφ and Ez of E inside and outside of the

dielectric interface at r = a must be the same, and similarly

for the tangential components Hφ and Hz.

• The boundary conditions are then given as:

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Cond…

• At, r = a;

Eφ1 = Eφ2

Ez1 = Ez2

Hφ1 = Hφ2

Hz1 = Hz2

• The boundary conditions give four equations in terms of

arbitrary constants, and the modal phase constant

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Modal Equation

• Consider the first tangential components of E, for the z

component we have, at inner core – cladding boundary (E

= Ez1) and at the outside of the boundary (E = Ez2), that

• The ϕ component is found from Maxwell’s Eq. inside the

core the factor q2 is given by

• where ,

• while outside the core

• With,

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1 2 ( ) ( ) 0z zE E AJ ua CK wa

2 2 2 2

1q u k

11 1

2 nk

2 2 2 2

2q w k

22 2

2 nk

Cond…

• So, we get

• Similarly, for tangential components of H it is readily

shown that, at r = a,

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1 2 2

2

( ) '( )

( ) '( ) 0

j jE E A J ua B uJ ua

u a

j jC K wa D wK wa

w a

1 2 12

12

( ) '( )

( ) '( ) 0

j jH H B J ua A uJ ua

u a

j jD K wa C wK wa

w a

Cond…

• are set of four equations with four unknown coefficients,

A, B, C and D. A solution to these equations exists only if

the determinant of these coefficients is zero, that is,

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1 1 1 1

2 2 2 20

3 3 3 3

4 4 4 4

A B C D

A B C D

A B C D

A B C D

Cond…

• A1 to A4 are coefficients of A

• B1 to B4 are coefficients of B

• C1 to C4 are coefficients of C

• D1 to D4 are coefficients of D

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Cond… 29 July 2012 UNIT 1

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2

1

1 ( )

2 ( )

3 0

4 '( )

A J ua

A J uaau

A

jA J ua

u

2

1 0

2 '( )

3 ( )

4 ( )

B

jB J ua

u

B J ua

B J uaau

2

2

1 ( )

2 ( )

3 0

4 '( )

C K wa

C K waaw

C

jC K wa

w

2

1 0

2 '( )

3 ( )

4 ( )

D

jD K wa

w

D K wa

D K waaw

Cond…

• Evaluation of the above determinant yields the following

eigenvalue equation for β

• Eq. is called characteristic equation.

• The characteristic equation contains three unknowns

namely .

• So using these Eq. we can find the modal propagation

constant .

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2 2

2 2

1 2 2 2

' ' ' ' 1 1J ua K wa J ua K wak k

uJ ua wK wa uJ ua wK wa a u w

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