DISPERSION & CUTOFF CHARACTERISTICS OF CIRCULAR …ajaykmga.weebly.com/uploads/2/2/1/5/2215854/m._tech_defence.pdf · important moments of making right decisions and I am glad to

THE DISSERTATION ENTITLED

DISPERSION & CUTOFF CHARACTERISTICS OF

CIRCULAR OPTICAL WAVEGUIDE

WITH HELICAL WINDING

Submitted in partial fulfillment of the requirements

For the degree of

Master of Technology

In

Electronics Engineering

With Specialization

In

Communication Systems

Submitted by

Ajay Kumar Gautam (P08EC901)

Under the Guidance of

Prof. B. R. Taunk

&

Dr. Vivekanand Mishra

JULY-2010

DEPARTMENT OF ELECTRONICS ENGINEERING

SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY

SURAT-395007

SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY DEPARTMENT OF ELECTRONICS ENGINEERING

SURAT-395007

CERTIFICATE

This is to certify that the thesis entitled “Dispersion & Cutoff Characteristics of

Circular Optical Waveguide with Helical Winding”, submitted by Ajay Kumar

Gautam (P08EC901) in the partial fulfillment of the requirements for award of Master in

Technology in Electronics Engineering with specialization in Communication Systems,

has satisfactorily presented during the year 2009-10.

External Examiner Internal Examiner Internal Examiner Chairman

Head of Dept. P. G. Incharge

Dr.(Ms) S. Patnaik Prof. J. N. Sarvaiya

SEAL OF DEPARTMENT

Acknowledgement

This project is by far the most significant accomplishment in my life and it would be

impossible without people who supported me and believed in me.

I would like to extend my gratitude and my sincere thanks to my honourable, esteemed

supervisor Prof. B. R. Taunk Sir and Dr. Vivekanand Mishra Sir, Department of

Electronics and Communication Engineering, S.V. NIT, Surat for their immeasurable

guidance and valuable time that he devoted for project. I sincerely thank for their

exemplary guidance and encouragement. Their trust and support inspired me in the most

important moments of making right decisions and I am glad to work with them.

I want to express great thanks to Dr. Vivek Singh, Professor, of Department of Physics of

Banaras Hindu University [BHU] Varanasi for providing a continuous motivation and

help and as well guiding me. He has been great sources of inspiration to me and I thank

them from the bottom of my heart.

I would also like to thanks our Head of Department Dr. (Mrs.) S. Patnaik and Prof. B. R.

Taunk (former HOD), ECED Department, S.V. NIT, Surat who provide me to all

facilities and coordination.

I would like to thank all my friends and especially my classmates for all the thoughtful

and mind stimulating discussions we had, which prompted us to think beyond the obvious.

I have enjoyed their companionship so much during my stay at S.V.NIT, Surat.

I would like to thank all those who made my stay in S.V.NIT, Surat an unforgettable and

rewarding experience.

AJAY KUMAR GAUTAM

Roll No. P08EC901

CONTENTS

List of Figures I

List of Tables. II

Abstract II

1. Introduction 1

1.1 Motivation 2

2. Optical Waveguide 3

2.1 Introduction

2.2 The optical fiber

2.3 The numerical aperture

2.4 Types of optical fiber

2.4.1 Single Mode Fibers

2.4.2 Multimode Fibers

2.5 Mode theory for circular optical waveguide

2.5.1 Maxwell’s Equations

2.5.2 Waveguide Equations

2.5.3 Wave Equations for Step – Index Fibers

2.5.4 Boundary Conditions

2.5.5 Modal Equation

2.5.6 Modes in Step – Index fibers

2.5.7 Cutoff conditions for fiber modes

3

4

6

7

8

9

9

12

13

16

23

23

26

27

3. Analysis of optical waveguide with helical winding 30

3.1 Helix

3.1.1 Types of Helix

3.2 Circular Optical Waveguide with conducting helical Winding

3.3 Boundary Conditions

3.4 Modal equation

30

30

32

32

33

4. Result and Discussion 38

4.1 Dispersion characteristics

4.1.1 Dispersion characteristics at pitch angle ψ = 00


4.1.3 Dispersion characteristics at pitch angle ψ =450

38

39

40

41



4.2 Dependence of cutoff values Vc

42

43

44

5. Conclusion & Future Work

5.1 Conclusion

5.2 Future Work

46

46

46

Reference 48

Publication 50

I

LIST OF FIGURES

Fig. 2.1 A typical optical fiber waveguide consists of thin cylindrical glass rod 3

Fig. 2.2 (a) Refractive index profile, multimode step - index fiber 4

(b) Refractive index profile, graded - index multi mode fiber 4

Fig. 2.3 A long optical fiber carrying a light beam 5

Fig. 2.4 Types of optical fiber 7

Fig. 2.5 Electric field distribution for several of lower - order guided modes 11

Fig. 2.6 Low-order and high-order modes 11

Fig. 2.7 Cylindrical coordinate system used for analyzing electromagnetic

wave propagation in an optical fiber 14

Fig. 2.8 Bessel functions of first kind 18

Fig. 2.9 Bessel functions of second kind 18

Fig. 2.10 Modified Bessel functions of first kind 19

Fig. 2.11 Modified Bessel functions of second kind 20

Fig. 2.12 Plots of the propagation constant b as a function of V for a lower –

order modes 29

Fig. 3.1 Example of helix as coil springs 30

Fig. 3.2 Helix (A) Right – handed, (B) Left – handed 31

Fig. 3.3 Fiber with circular cross – section wrapped with a sheath helix 32

Fig. 4.1 Dispersion Curve for pitch angle ψ = 00 39





Fig. 4.6 Dependence of cutoff values Vc on the pitch angle ψ 45

II

LIST OF TABLES

Table 2.1 Cutoff conditions for some lower – order modes 28

Table 4.1 Cutoff Vc values for some lower – order modes 45

III

ABSTRACT

The objective of this thesis is to study the properties of circular optical waveguide

using Bessel function and to measure the dispersion characteristics using the helical

windings at core-cladding interface. Then after, we have used helical windings to

study the performance characteristics of waveguide with helical windings on

dielectric material. Once the properties of helical windings have been evaluated, then

we study the performance of this characteristic at different pitch angles. Boundary

conditions have been used to obtain the dispersion characteristics and these conditions

have been utilized to get the model Eigen values equation. From these Eigen value

equations dispersion curve are obtained and plotted for modified optical waveguide

for particular values of the pitch angle of the winding and the result has been

compared.

Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding

Sardar Vallabhbhai National Institute of Technology, Surat Page 1

Chapter 1

Introduction

An optical waveguide is basically a cylindrical dielectric waveguide with a circular cross

section where a high-index wave guiding core is surrounded by a low-index cladding. The

index step and profile are controlled by the concentration and distribution of dopants. For

example, the core can be doped with Germania (GeO2) or alumina (Al2O3) or other

oxides, such as P2O5 or TiO2, for a slightly higher index than that of a silica cladding [1].

Silica fibers are ideal for light transmission in the visible and near-infrared regions

because of their low loss and low dispersion in these spectral regions. They are therefore

suitable for optical communications. Even though optical fiber seems quite flexible, it is

made of glass, which cannot withstand sharp bending or longitudinal stress. Therefore

when fiber is placed inside complete cables special construction techniques are employed

to allow the fiber to move freely within a tube. Usually fiber optic cables contain several

fibers, a strong central strength member and one or more metal sheaths for mechanical

protection. Some cables also include copper pairs for auxiliary applications.

Optical fibers are manufactured in three main types: multi-mode step-index, multi-mode

graded-index, and single – mode. Multi – mode step – index fiber has the largest

diameter core (typically 50 to 100 um) [2]. The larger distance between interfaces allows

the light rays to travel the most distance when bouncing through the cable. Multi – mode

fibers typically carry signals with wavelengths of 850 nm or 1300 nm.

Optical fibers allow data signals to propagate through them by ensuring that the light

signal enters the fiber at an angle greater than the critical angle of the interface between

two types of glass. To use fiber optic cables for communications, electrical signals must

be converted to light [2], transmitted, received, and converted back from light to electrical

signals. This requires optical sources and detectors that can operate at the data rates of the

communications system.

With the cost of optical fiber technology continuing to decrease, many of today’s

businesses are utilizing this technology in building distribution and/or workstation



applications. Optical fiber’s inherent immunity to both electromagnetic interference

(EMI) and radio interference (RFI), and its relatively light weight and enormous

bandwidth capabilities make it ideal for voice, video and high-speed data applications.

Optical fibers have a wide range of applications [1]. Owing to their low losses and large

bandwidths, their most important applications are fiber-optic communications and

interconnections. Other important applications include fiber sensors, guided optical

imaging, remote monitoring, and medical applications.

Optical fibers with helical winding are known as complex optical waveguides. The use of

helical winding in optical fibers makes the analysis much accurate. As the number of

propagating modes depends on the helix pitch angle, so helical winding at core – cladding

interface can control the dispersion characteristics of the optical waveguide [3].

1.1 Motivation

The conventional optical fiber having a circular core cross – section which is widely used

in optical communication systems [1]. Recently metal – clad optical waveguides have

been studied because these provide potential applications, connecting the optical

components to other circuits. Metallic – cladding structure on an optical waveguide is

known as a TE – mode pass polarizer and is commercially applied to various optical

devices [4]. The propagation characteristics of optical fibers with elliptic cross – section

have been investigated by many researchers. Singh [5] have proposed an analytical study

of dispersion characteristics of helically cladded step – index optical fiber with circular

core. The model characteristic and dispersion curves of a hypocycloidal optical

waveguide have been investigated by Ojha [6]. Present work is the study of circular

optical waveguide with sheath helix [3] in between the core and cladding region, this

work also gives the comparison of dispersion characteristic at different pitch angles. The

sheath helix is a cylindrical surface with high conductivity in a preferential direction

which winds helically at constant angle around the core – cladding boundary surfaces.



Chapter 2

Optical Waveguide

2.1 Introduction

An optical waveguide is a structure which confines and guides the light beam by the

process of total internal reflection. The most extensively used optical waveguide is the

step index optical waveguide which consists of a cylindrical central core, clad by a

material of slightly lower refractive index. If the refractive indices of the core and

cladding are n1 and n2 respectively, then for a ray entering the fiber, if the angle of

incident (at the core – cladding interface) θA is greater than the critical angle.

–1

C 2 1 sin n / n , (2.1)

Fig 2.1 A typical optical fiber waveguide consists of thin cylindrical glass rod [8]

then the ray will undergo total internal reflection at that interface. Furthermore, because

of the cylindrical symmetry in the fiber structure, this ray will suffer total reflection at the

lower interface also and will therefore be guided through the core by repeated total

internal reflections. This is the basic principle of the light guidance through the optical

fiber.

The simplest optical waveguide is the planner waveguide which consists of a thin

dielectric film (of refractive index n1) sandwiched between materials of slightly lower

refractive indices. Although all waveguides used in integrated optics are asymmetric in

nature, the electromagnetic analysis of a symmetric waveguides is much easier to



understand and at same time it brings out almost all the salient points associated with the

modes of a waveguide, therefore making it easier to understand the physical principles of

more complicated guiding structures [7].

2.2 The optical fiber

Fig 2.1 shows a glass fiber which consists of a (cylindrical) central core cladded by a

material of slightly lower refractive index. The corresponding refractive index

distribution is given by as shown in Fig. 2.2.

1

2

;( )

;

n r an r

n r a

, (2.2)

For the ray entering the fiber, if the angle of incidence (at the core – cladding interface)

θA is than the critical angle θC, then the ray will undergo total internal reflection at that

interface.

(a) (b)

Fig. 2.2 (a) Refractive index profile, multimode step - index fiber,

(b) Refractive index profile, graded - index multi mode fiber [9]



Furthermore, because of the cylindrical symmetry in the fiber structure, this ray will

suffer total internal reflection at the lower interface also and will therefore be guided

through the core by repeated total internal reflections.

Fig. 2.3 A long optical fiber carrying a light beam [10]

Fig. 2.3 shows the actual guidance of a light beam as it propagates through a long optical

fiber. It is necessary to use a cladded fiber (Fig. 2.1) rather than a bare fiber because of

the fact that for transmission of light from one place to another, the fiber must be

supported, and the supporting structures may considerably distort the fiber thereby

affecting the guidance of the light wave.

This can be avoided by choosing a sufficiently thick cladding. Further, in a fiber bundle,

in the absence of the cladding, light can leak from one fiber to another.



2.3 The numerical aperture

Consider a ray which is incident on the entrance aperture of the fiber making an angle θA

with the axis. Let the reflected ray make an angle θR with the axis. Assuming the outside

medium to have a refractive index n0 (which for most practical cases is unity), we get

1

0

sin

sin

A

R

n

n

(2.3)

Obviously if this ray has to suffer total internal reflection at the core – cladding interface,

sin θR (= cos θR) > 2

1

n

n

Thus sin

12 2

2R

1

n 1 –

n

And we must have

12

2 2 21 2 1 2

2

0 1 0

sin 1A

n n n n

n n n

If (n12 – n2

2) n0

2 then for all values of θA, total internal reflection will occur. Assuming

n0 = 1, the maximum value of sin θA for a ray to be guided is given by

1

2 2 2 221 2 1 2

,max2 2

1 2

( ) ; , 1sin

1; , 1A

n n when n n

when n n

(2.4)

Thus, if a cone of light is incident on one end of fiber, it will be guided through it

provided the semi angle of the cone is less than ,maxA . This angle is a measure of the light

gathering power of the fiber and as such, one defines the numerical aperture (NA) of the

fiber the following equation

1 1

2 2 2 21 2 1( ) (2 )NA n n n (2.5)



where,

2 2

1 2 1 2

2

1 12

n n n n

n n

(2.6)

Also 1 2n n this is indeed the case for all practical optical fibers.

2.4 Types of optical fiber

Optical fibers are characterized by their structure and by their properties of transmission.

Basically, optical fibers are classified into two types. The first type is single mode fibers.

The second type is multimode fibers. As each name implies, optical fibers are classified

by the number of modes that propagate along the fiber. As previously explained, the

structure of the fiber can permit or restrict modes from propagating in a fiber. The basic

structural difference is the core size. Single mode fibers are manufactured with the same

materials as multimode fibers. Single mode fibers are also manufactured by following the

same fabrication process as multimode fibers.

Fig. 2.4 Types of optical fiber [12]



2.4.1 Single Mode Fibers

The core size of single mode fibers is small. The core size (diameter) is typically around 6

to 10 m. A fiber core of this size allows only the fundamental or lowest order mode to

propagate around a 1300 nanometer (nm) wavelength. Single mode fibers propagate only

one mode, because the core size approaches the operational wavelength. The value of the

normalized frequency parameter (V) relates core size with mode propagation.

In single mode fibers, V is less than or equal to 2.405. When V is 2.405, single mode

fibers propagate the fundamental mode down the fiber core, while high-order modes are

lost in the cladding. For low V values, most of the power is propagated in the cladding

material. Power transmitted by the cladding is easily lost at fiber bends. The value of V

should remain near the 2.405 level.

1 1

2 2 2 21 2 1

0 0

2 2( ) (2 )V a n n an

(2.7)

where “V” is known as waveguide parameter, V number or V parameter. Practical single

mode fibers have varying from 0.2% to 0.5% and typical core diameters in the range

10 – 6 m.

Single mode fibers have a lower signal loss and a higher information capacity

(bandwidth) than multimode fibers. Single mode fibers are capable of transferring higher

amounts of data due to low fiber dispersion. Basically, dispersion is the spreading of light

as light propagates along a fiber. Dispersion mechanisms in single mode fibers are

discussed in more detail later in this chapter. Signal loss depends on the operational

wavelength. In single mode fibers, the wavelength can increase or decrease the losses

caused by fiber bending. Single mode fibers operating at wavelengths larger than the

cutoff wavelength lose more power at fiber bends. They lose power because light radiates

into the cladding, which is lost at fiber bends. In general, single mode fibers are

considered to be low-loss fibers, which increase system bandwidth and length.



2.4.2 Multimode Fibers

As their name implies, multimode fibers propagate more than one mode. Multimode

fibers can propagate over 100 modes. The number of modes propagated depends on the

core size and numerical aperture (NA). As the core size and

NA increase, the number of modes increases. Typical values of fiber core size and NA are

50 to 100 m and 0.20 to 0.29, respectively.

A large core size and a higher NA have several advantages. Light is launched into a

multimode fiber with more ease. The higher NA and the larger core size make it easier to

make fiber connections. During fiber splicing, core-to-core alignment becomes less

critical. Another advantage is that multimode fibers permit the use of light-emitting

diodes (LEDs). Single mode fibers typically must use laser diodes. LEDs are cheaper, less

complex, and last longer. LEDs are preferred for most applications.

Multimode fibers also have some disadvantages. As the number of modes increases, the

effect of modal dispersion increases. Modal dispersion (intermodal dispersion) means that

modes arrive at the fiber end at slightly different times. This time difference causes the

light pulse to spread. Modal dispersion affects system bandwidth. Fiber manufacturers

adjust the core diameter, NA, and index profile properties of multimode fibers to

maximize system bandwidth.

2.5 Mode theory for circular optical waveguide

The optical waveguide is the fundamental element that interconnects the various devices

of an optical integrated circuit, just as a metallic strip does in an electrical integrated

circuit. However, unlike electrical current that flows through a metal strip according to

Ohm’s law, optical waves travel in the waveguide in distinct optical modes. 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. The mode theory is used to describe the properties of light that ray

theory is unable to explain. The mode theory uses electromagnetic wave behavior to



describe the propagation of light along a fiber. A set of guided electromagnetic waves is

called the modes of the fiber [13, 14, 25].

For a given mode, a change in wavelength can prevent the mode from propagating along

the fiber. The mode is no longer bound to the fiber. The mode is said to be cut off. Modes

that are bound at one wavelength may not exist at longer wavelengths. The wavelength at

which a mode ceases to be bound is called the cutoff wavelength for that mode.

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.

The wavelength that prevents the next higher mode from propagating is called the cutoff

wavelength of the fiber. An optical fiber that operates above the cutoff wavelength (at a

longer wavelength) is called a single mode fiber. An optical fiber that operates below the

cutoff wavelength is called a multimode fiber.

In a fiber, the propagation constant of a plane wave is a function of the wave's wavelength

and mode. The change in the propagation constant for different waves is

called dispersion. The change in the propagation constant for different wavelengths is

called chromatic dispersion. The change in propagation constant for different modes is

called modal dispersion.

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. The transverse modes,

shown in Fig. 2.5, propagate along the axis of the fiber. The mode field patterns shown in

Fig. 2.5 are said to be transverse electric (TE). In TE modes, the electric field is

perpendicular to the direction of propagation.

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 electric field is in

the direction of propagation. Fig. 2.5 shows only TE modes.



Fig. 2.5 Electric field distribution for several of lower - order guided modes [13]

The TE mode field patterns shown in Fig. 2.5 indicate the order of each mode. 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. Generally, modes with more than a few (5-10)

field maxima are referred to as high-order modes [13].

The order of the mode is also determined by the angle the wavefront makes with the axis

of the fiber. Fig. 2.6 illustrates light rays as they travel down the fiber. These light rays

indicate the direction of the wavefronts. High-order modes cross the axis of the fiber at

steeper angles. Low-order and high-order modes are shown in Fig. 2.6.

Fig. 2.6 Low-order and high-order modes [13]



Notice that the modes are not confined to the core of the fiber. The modes extend partially

into the cladding material. Low-order modes penetrate the cladding only slightly. In low-

order modes, the electric and magnetic fields are concentrated near the center of the fiber.

However, high-order modes penetrate further into the cladding material. In high-order

modes, the electrical and magnetic fields are distributed more toward the outer edges of

the fiber.

This penetration of low-order and high-order modes into the cladding region indicates

that some portion is refracted out of the core. The refracted modes may become trapped in

the cladding due to the dimension of the cladding region. The modes trapped in the

cladding region are called cladding modes. As the core and the cladding modes travel

along the fiber, mode coupling occurs. Mode coupling is the exchange of power between

two modes. Mode coupling to the cladding results in the loss of power from the core

modes.

2.5.1 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

BX E

t

(2.8a)

DX H

t

(2.8b)

. 0D (2.8c)

. 0B (2.8d)

Where D = E and B = µH. The parameter is the permittivity (or dielectric constant) and

µ is the permeability of the medium.

A relationship defining the wave phenomena of the electromagnetic fields can be derived

from Maxwell’s equations [14]. Taking the curl of Eq. (2.8a) and making use of Eq.

(2.8b) yields



2

2( ) ( )

EX X E X H

t t

(2.9a)

Using the vector identity

2( ) ( . )X X E E E

And using Eq. (2.8c), Eq. (2.9a) becomes

22

2

EE

t

(2.9b)

Similarly, by taking the curl of Eq. (2.8b), it can be shown that

22

2

HH

t

(2.9c)

Equations (2.9b) and (2.9c) are the standard wave equations [14, 25].

2.5.2 Waveguide Equations

Consider electromagnetic waves propagating along the cylindrical fiber shown in Fig. 2.7.

For this fiber, a cylindrical coordinates system ( , , )r z is defined with the z axis lying of

the waveguide. If the electromagnetic waves are to propagate along the z axis, they will

have a functional dependence of the form

( )

0( , ) j t zE E r e (2.10a)

( )

0( , ) j t zH H r e (2.10b)

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.

An optical mode refers to a specific solution of the wave equation that satisfies the

appropriate boundary conditions and has the property that its spatial distribution does not

change with propagation [14, 25].



When Eq. (2.10a) and (2.10b) are substituted into Maxwell’s curl equations, we have,

from Eq. (2.8a)

1 zr

Ejr E j H

r

(2.11a)

zr

Ej E j H

r

(2.11b)

1 r

z

ErE j H

r r

(2.11c)

And, from Eq. (2.8b),

1 zr

Hjr H j E

r

(2.12a)

zr

Hj H j E

r

(2.12b)

1 r

z

HrH j E

r r

(2.12c)

Fig. 2.7 Cylindrical coordinate system used for analyzing electromagnetic

wave propagationin an optical fiber [14]

By eliminating variables these equations can be rewritten such that, when Ez and Hz are

known, the remaining transverse components Er, Eϕ, Hr and Hϕ can be determined. For



example Eϕ or Hr can be eliminated from Eq. (2.11a) and Eq. (2.12b) so that the

component Hϕ or Er, respectively, can be found in terms of Ez or Hz. These yields

2

z zr

E HjE

q r r

(2.13a)

2

z zE HjE

q r r

(2.13b)

2

z zr

H EjH

q r r

(2.13c)

2

z zH EjH

q r r

(2.13d)

where q2 =

2 - β

2 = k

2 - β

2.

Substitution of Eq. (2.13c) and Eq. (2.13d) into Eq. (2.12c) gives the wave equation [14 -

16] in cylindrical coordinates,

2 2 22

2 2 2 2

1 10z z z z

z

E E E Eq E

r r r r z

(2.14)

And substitution of Eq. (2.13a) and Eq. (2.13b) into Eq. (2.11c) gives,

2 2 22

2 2 2 2

1 10z z z z

z

H H H Hq H

r r r r z

(2.15)

It is interesting to note that Eq. (2.14) and Eq. (2.15) each contain either only Ez or only

Hz. This implies that the longitudinal components of E and H are uncoupled and can be

chosen arbitrary provided that they satisfy Eq. (2.14) and Eq. (2.15). However the

coupling of Ez and Hz is required by the boundary conditions of the electromagnetic field

components. If the boundary conditions do not lead to coupling between the field

components, 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 designed 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.

2.5.3 Wave Equations for Step – Index Fibers

We can use Eq. (2.14) and Eq. (2.15) to find the guided modes in a step – index fiber. A

standard mathematical procedure for solving equations such as Eq. (2.14) is to use the

separation – of – variables method, which assumes a solution of the form

Ez = AF1(r) F2(ϕ) F3(z) F4(t) (2.16)

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

F3(z)F4(t) =ej(ωt−βz)

(2.17)

Since the wave is sinusoidal in time and propagates in the z direction. Also because of the

symmetry of the waveguide, each field component must not change when the coordinate

ϕ is increased by 2π. We thus assume a periodic function of the form

2 ( ) jF e (2.18)

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. (2.17) and Eq. (2.18) into Eq. (2.16), the wave equation for Ez [Eq.

(2.14)] becomes

2 221 1

12 2

10

F Fq F

r r r r

(2.19)

This is well - known differential equation for Bessel functions [15]. An exactly identical

equation can be derived for Hz.

Consider a homogeneous core of refractive index n1 and radius a, which is surrounded by

an infinite cladding of index n2. The reason for assuming an infinitely thick cladding is

that the guided modes in the core have exponentially decaying fields outside the core and

these must have insignificant values at the outer boundary of the cladding. In practice,



optical fiber designed with claddings that are sufficiently thick so that the guided – mode

field does not reach the outer boundary of the cladding. To get an idea of field patterns,

the electric field distributions for several of the lower guided modes in a symmetrical –

slab were shown in Fig. 2.6.

The fields vary harmonically in the guiding region of refractive index n1 and decay

exponentially outside of this region.

Eq. (2.19) must now be solved for the regions inside and outside the core. For the inside

region the solutions for the guided modes must remain finite as r 0, whereas on the

outside the solutions must decay to zero as r . As Eq. (2.19) is standard differential

equation for Bessel function, so we must take solution in the form of Bessel function.

But first we have to choose appropriate Bessel function for solution of Eq. (2.19) We

have a variety of solutions to the Bessel’s equation depending upon the parameters

and q . is an integer and a positive quantity.

Depending upon the choice of q i.e., a) real, b) imaginary, c) complex, we get different

solutions to the Bessel’s equation. So to choose the proper solution, let us now look at the

plot of the Bessel functions for various possibilities of q (argument). There are three

different types of Bessel functions depending upon the nature of q .

Let us now look at the plot of the Bessel functions for various possibilities of

q (argument). There are three different types of Bessel functions depending upon the

nature of q .

If q is real then the solutions are

J qr Bessel functions of first kind

( )Y qr Bessel functions of second kind

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

function. Plots of the two functions as a function of their arguments are shown in the Fig.

2.8 and Fig. 2.9.



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. 2.8 Bessel functions of first kind [15]

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. 2.9 Bessel functions of second kind [15]

We can see from Fig. 2.8 that except 0J , all the other Bessel functions of first kind go to

zero as the argument goes to zero. Only 0J approaches 1 as its argument approaches



zero. All Bessel functions of first kind have oscillatory behavior and their amplitude

slowly decreases as the argument increases.

Fig. 2.9 shows the behavior of the Neumann function as a function of its argument, qr .

The important thing to note is, the Bessel functions of first kind J are finite for all

values of the argument, whereas the Bessel functions of second kind are finite for all

values of argument except zero. When the argument tends to zero, the Bessel functions of

second kind tend to .

If q is imaginary, we get solutions of the Bessel’s equation as

/I qr j Modified Bessel functions of first kind

( / )K qr j Modified Bessel functions of second kind

Since q is imaginary, ( / )qr j is a real quantity. So the argument of the modified Bessel

functions is real.

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. 2.10 Modified Bessel functions of first kind [15]



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. 2.11 Modified Bessel functions of second kind [15]

The modified Bessel functions are shown in the Fig. 2.10 and Fig. 2.11. The I functions

are monotonically increasing functions of /qr j , and K functions are monotonically

decreasing functions of ( / )qr j .

If q is complex, then the solutions are

Hankel functions of first kind

Hankel functions of second kind

But as our medium is lossless, in this case q can either be real or imaginary, so no need to

study the case when the q is complex.

Now 2 2 2q , where is the propagation constant of the wave along the

z direction. If we assume the situation is lossless i.e. when the wave travels in the

z direction, its amplitude does not change as a function of z , then should be a real

quantity. If become imaginary, the function j ze becomes an exponentially decaying

function, and there is no wave propagation. For wave propagation inside an optical fiber

(1)H qr

(2)H qr



we assume that the material is lossless. Then the dielectric constant is a real quantity.

This makes 2 a real quantity. Also for a propagating mode 2 is a real quantity.

Hence, 2 2 2q is also a real quantity albeit it can be positive or negative. In

other words, q can be real or imaginary depending upon whether 2 is greater or

lesser than 2 . For a lossless case, we have a solution which is a linear combination of

Bessel functions of first kind and modified Bessel functions of second kind.

As far as guided wave propagation is concerned, the fields should have oscillatory

behavior inside the core, and in cladding the field must decay monotonically. Therefore it

is obvious that inside the core the Modified Bessel function is not the proper solution.

Only Bessel function of first kind could be solutions inside the core.

Let us now re-look at the two functions, Bessel functions of first kind (Fig. 2.8) and

Bessel functions of second kind (Fig. 2.9), and make following observations.

Bessel functions of first kind: The functions uJ r are finite for all values of r .

Bessel functions of second kind: The functions ( )Y ur start from at 0r and have

finite value for all other values of r.

For the core 0r represents the axis of fiber. Therefore if a Bessel function of second

kind is chosen as a solution, the field strength would be at the axis of the fiber which is

inconsistent with the physical conditions. The fields must be finite all over the cross

section of the core. So the Bessel function of second kind cannot be the solution if 0r

point is included in the region under consideration.

Therefore we can conclude that only uJ r is the appropriate solution for the

modal fields inside the core of an optical fiber.

Let us now look at the modified Bessel’s functions, as shown in figures. For modified

Bessel’s functions of the 1st kind (Fig. 2.10), as r increases, that is, as we move away

from the axis of the fiber the field monotonically increases and when r field goes to

infinity. Since the energy source is inside the core, the fields cannot grow indefinitely



away from the core. The only acceptable situation is that the field decays away from the

core i.e., for larger values of r . This behavior is correctly given by the Modified Bessel

function of second kind, ( )K wr (Fig. 2.11).

So we can conclude that the modified Bessel function of 1st kind ( )I wr is not

appropriate solution in the cladding. The correct solution would be only Modified

Bessel function of 2nd

kind, ( )K wr .

In all then, the fields inside the core are given by ( )J ur and in the cladding are given by

( )K wr .

Thus for r < a, the solutions are Bessel function of first kind of order . For these

functions we use the common designation J ur . Here, 2 2 2

1u k with 11

2 nk

.

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

1 ( ) j j z j t

zE AJ ua e

(2.20)

1

j j z j t

zH BJ ua e

(2.21)

Outside of the core the solutions of Eq. 2.19 are given by modified Bessel functions of the

second kind, ( )K wa , where 2 2 2

2w k with 22

2 nk

.

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

2

j j z j t

zE CK wa e

(2.22)

2

j j z j t

zH DK wa e

(2.23)

where , , ,A B C D are arbitrary constants which are to be evaluated from the boundary

conditions. Also J ua and ( )K wa are the Bessel functions.

For a guided mode, the propagation constant lies between two limits 2 and 1 . If

2 2 1 1n k k k n k 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 2

k

is free – space propagation

constant.

2.5.4 Boundary Conditions

The solutions for β must be determined from the boundary conditions [14, 17]. The

boundary conditions requires that the tangential components E and zE of E inside and

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

tangential components H and zH .

The boundary conditions are then given as:

At r a ,

1 2E E (2.24a)

1 2z zE E (2.24b)

1 2H H (2.24c)

1 2z zH H (2.24d)

The boundary conditions give four equations in terms of arbitrary constants, , , ,A B C D

and the modal phase constant .

2.5.5 Modal Equation

Consider the first tangential components of E, for the z component we have, from Eq.

(2.20) at inner core – cladding boundary (E = Ez1) and from Eq. (2.22) at the outside of

the boundary (E = Ez2), that

1 2 ( ) ( ) 0z zE E AJ ua CK wa (2.25)

The ϕ component is found from Eq. (2.13b) inside the core the factor q2

is given by

2 2 2 2

1q u k (2.26)



where 11 1

2 nk

, while outside the core

2 2 2 2

2q w k (2.27)

with 22 2

2 nk

. Substituting Eq. (2.20) and Eq. (2.21) into Eq. (2.13b) to find

Eϕ1.

Similarly, using Eq. (2.22) and Eq. (2.23) to determine Eϕ2, yields, at r = a,

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

(2.28)

where the prime indicates differentiation with respect to the argument.

Similarly, for tangential components of H it is readily shown that, at r = a,

1 2 ( ) ( ) 0z zH H BJ ua DK wa (2.29)

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

(2.30)

Eq. (2.25), Eq. (2.28), Eq. (2.29) and Eq. (2.30) 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,

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

(2.31)

where,



A1 to A4 are coefficients of A in Eq. (2.25), Eq. (2.28), Eq. (2.29) and Eq. (2.30).

B1 to B4 are coefficients of B in Eq. (2.25), Eq. (2.28), Eq. (2.29) and Eq. (2.30).

C1 to C4 are coefficients of C in Eq. (2.25), Eq. (2.28), Eq. (2.29) and Eq. (2.30).

D1 to D4 are coefficients of D in Eq. (2.25), Eq. (2.28), Eq. (2.29) and Eq. (2.30).

Also,

2

1

1 ( )

2 ( )

3 0

4 '( )

A J ua

A J uaau

A

jA J ua

u

(2.32a)

2

1 0

2 '( )

3 ( )

4 ( )

B

jB J ua

u

B J ua

B J uaau

(2.32b)

2

2

1 ( )

2 ( )

3 0

4 '( )

C K wa

C K waaw

C

jC K wa

w

(2.32c)

2

1 0

2 '( )

3 ( )

4 ( )

D

jD K wa

w

D K wa

D K waaw

(2.32d)

Evaluation of the above determinant yields the following eigenvalue equation for β.

(2.33)



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

Eq. (2.33) is called characteristic equation [14, 25]. The characteristic equation contains

three unknowns namely , ,u w . So using the Eq. (2.26), Eq. (2.27) & Eq. (2.33) we can

find the modal propagation constant .

2.5.6 Modes in Step – Index fibers

Eq. (2.33) is called the characteristic equation. Its contain J - type Bessel functions. The J

- type Bessel functions are similar to harmonic functions since they exhibit oscillatory

behavior for real k, as is the case for sinusoidal functions. Because of the oscillatory

behavior of J , there will be m roots of Eq. (2.33) for a given value. These roots are

designated by m , and the corresponding modes [14, 25] are either , ,m mTE TM

m mEH or HE . If we take 0zH , all field components are expressed in terms of zE and

whatever fields we get, they do not have any magnetic field component in the direction of

propagation. We call this mode the Transverse Magnetic mode (TM mode). Similarly

if 0zE , the mode is called the Transverse Electric mode (TE mode). If both the

longitudinal components of the fields ( zE and zH ) are non-zero then we call the mode

the Hybrid mode. This mode is a combination of TE and TM modes. For a hybrid mode,

if we calculate the contribution by zE and zH to the transverse fields, one of them i.e. zE

or zH would dominate. Depending upon which of them contributes more, we can sub-

classify the Hybrid modes. If zE Dominates EH mode If zH DominatesHE mode.

Each of the above modes is characterized by two indices, and m (solution number).

The mode are therefore designated as ,m mTE TM & ,m mEH HE .

For the dielectric fiber waveguide, all modes are hybrid modes except those which 0 .

When 0 the right – hand side of Eq. (2.33) vanishes and two different equations

result. These are

0 0

0 0

' '0

J ua K wa

uJ ua wK wa

(2.34a)



0 02 2

1 2

0 0

' '0

J ua K wak k

uJ ua wK wa

(2.34b)

Using recurrence relation for the Bessel function, we have 0 1'( ) ( )J x J x and

0 1'( ) ( )K x K x . Put these recurrence relation for the Bessel function in Eq. (2.34a) and

Eq. (2.34b), we get

1 1

0 0

0J ua K wa

uJ ua wK wa

(2.35a)

Which corresponds to TE0m modes (Ez = 0), and

1 12 2

1 2

0 0

0J ua K wa

k kuJ ua wK wa

(2.35b)

Which corresponds to TM0m modes (Hz = 0).

When 0 the situation is more complex and numerical methods are needed to solve Eq.

(2.33) exactly.

2.5.7 Cutoff conditions for fiber modes

The cutoff condition [14] is the point at which a mode is no longer bound to the core

region. So its field no longer decays on the outside of the core. The cutoffs for the various

modes can be found by solving Eq. (2.33) in the limit 2 0w . This is, in general, fairly

complex, so that only the results, which are listed in Table 2.1.



Table 2.1 Cutoff conditions for some lower – order modes [14]

Mode Cutoff condition

0 TE0m, TM0m 0 ( ) 0J ua

1 HE1m, EH1m 1( ) 0J ua

2

mEH

mHE

( ) 0J ua

2

112

2

1 ( ) ( )1

n uaJ ua J ua

n

The permissible range of β for bound solutions is therefore

2 2 1 1n k k k n k (2.36)

Where 2 /k is the free – space propagation constant.

An important parameter connected with the cutoff condition is the normalized frequency

V (also called the V number or V parameter) [14] defined by

2 2

2 2 2 2 2 2 2

1 2

2 2( ) ( )

a aV u w a n n NA

(2.37)

which is dimensionless number that determines how many modes a fiber can support. The

number of modes that can exist in a wave guide as a function of V may be conveniently

represented in terms of a normalized propagation constant b [14] defined by

2 2 2

2

2 2

1 2

( / )k nawb

V n n

(2.38)

A plot of b as function of V is shown in Fig. 2.13 for few of the lower – order modes.



Fig. 2.12 Plots of the propagation constant b as a function of V for a lower – order modes [25]

This figure shows that each mode can exist only for values of V that exceed a certain

limiting value. The modes are cutoff when β/k = n2. The HE11 mode has no cutoff and

ceases to exist only when the core diameter is zero. This is the principle on which the

single mode fiber is based. By appropriately choosing a, n1 and n2 so that

2 2 1/2

1 2

2( ) 2.405

aV n n

(2.39)

Which is the value at which the lowest – order Bessel function J0 = 0, all modes except

the HE11 mode are cutoff.



Chapter 3

Analysis of optical waveguide with helical winding

3.1 Helix

A helix [18] is a type of space curve, i.e. a smooth curve in three - dimensional space. It

is characterized by the fact that the tangent line at any point makes a constant angle with a

fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral

staircases (Fig. 3.1).

Fig. 3.1 Example of helix as coil springs [19]

3.1.1 Types of Helix

Helices can be either right-handed or left-handed. With the line of sight along the helix's

axis, if a clockwise screwing motion moves the helix away from the observer, then it is

called a right-handed helix (Fig. 3.2); if towards the observer then it is a left-handed helix.

Handedness (or chirality) is a property of the helix, not of the perspective: a right-handed

helix cannot be turned or flipped to look like a left-handed one unless it is viewed in a



mirror, and vice versa. Most hardware screws are right-handed helices.

Fig. 3.2 Helix (A) Right – handed, (B) Left – handed [20]

The pitch of a helix is the width of one complete helix turn, measured parallel to the axis

of the helix. A double helix consists of two (typically congruent) helices with the same

axis, differing by a translation along the axis, which may or may not measure half the

pitch.

A conic helix may be defined as a spiral on a conic surface, with the distance to the apex

an exponential function of the angle indicating direction from the axis. An example is the

Corkscrew [21] roller coaster at Cedar Point amusement park.

A circular helix has constant band curvature and constant torsion. A curve is called a

general helix or cylindrical helix if its tangent makes a constant angle with a fixed line

in space. A curve is a general helix if and only if the ratio of curvature to torsion [22] is

constant.

A sheath helix [24] can be approximated by winding a very thin conducting wire around

the cylindrical surface so that the spacing between the adjacent windings is very small

and yet they are insulated from each other as shown in Fig. 3.3.



3.2 Circular Optical Waveguide with conducting helical Winding

We consider the case of a fiber with circular cross – section wrapped with a sheath helix

at core – clad boundary as shown in Fig. 3.3.

Fig 3.3 Fiber with circular cross – section wrapped with a sheath helix

In our structure, the helical windings are made at a constant angle ψ – the helix pitch

angle. The structure has high conductivity in a preferential direction. The pitch angle can

control the propagation behavior of such fibers [23]. We assume that the core and

cladding regions have the real refractive indices n1 and n2 (n1 > n2), and (n1-n2) / n1 << 1.

The winding is right – handed and the direction of propagation is positive z direction. The

winding angle of the helix (pitch angle - ψ) can take any arbitrary value between 0 to π/2.

This type of fibers is referred to as Circular helically cladded fiber (CHCF). This analysis

requires the use of cylindrical coordinate system ( , , )r z [24] with the z – axis being the

direction of propagation.

3.3 Boundary Conditions

Tangential component of the electric field in the direction of the conducting winding

should be zero, and in the direction perpendicular to the helical winding, the tangential

component of both the electric field and magnetic field must be continuous, so we have

following boundary condition [17] with helix.



1 1 0zE sin E cos (3.1a)

2 2 0zE sin E cos (3.1b)

1 2 1 2 0z zE E cos E E sin (3.1c)

1 2 1 2 0z zH H sin H H cos (3.1d)

3.4 Model equation

The guided mode along this type of fiber can be analyzed in a standard way, with the

cylindrical coordinates system ( , , )r z . In order to have a guided field the following

conditions must be satisfied 2 2 1 1n k k k n k , where n1 and n2 are refractive

indices or core and cladding regions respectively. The solution of the axial field

components can be written as,


1 ( ) j j z j t

zE AJ ua e

(3.2a)

1

j j z j t

zH BJ ua e

(3.2b)


2

j j z j t

zE CK wa e

(3.2c)

2

j j z j t

zH DK wa e

(3.2d)

where , , ,A B C D are arbitrary constants which are to be evaluated from the boundary

conditions. Also J ua and ( )K wa are the Bessel functions.

For a guided mode, the propagation constant lies between two limits 2 and 1 . If

2 2 1 1n k k k n k 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 2

k

is free – space propagation

constant.



The transverse field components can be obtained by using Maxwell’s standard relations.

So the electric and magnetic field components Eϕ and Hϕ can be written as,

The expressions for Eϕ and Hϕ inside the core are, when (r < a)

1 2( ) '( ) j j z j tj

E j AJ ua uBJ ua eu a

(3.3a)

1 12( ) '( ) j j z j tj

H j BJ ua uAJ ua eu a

(3.3b)

The expressions for Eϕ and Hϕ outside the core are, when (r > a)

2 2( ) '( ) j j z j tj

E j CK wa wDK wa ew a

(3.4a)

2 22( ) '( ) j j z j tj

H j DK wa wCK wa ew a

(3.4b)

Now put these transverse field components equations into boundary conditions, we get

following four unknown equations involving four unknown arbitrary constants

2( ) sin cos '( ) cos 0

jAJ ua BJ ua

u a u

(3.4a)

2( ) sin cos '( ) cos 0

jCK wa DK wa

w a w

(3.4b)

2

2

( ) cos sin '( ) sin

( ) cos sin '( ) sin 0

jAJ ua BJ ua

u a u

jCK wa DK wa

w a w

(3.4c)

1

2

2

2

'( ) cos ( ) sin cos

'( ) cos ( ) sin cos 0

jAJ ua BJ ua

u u a

jCK wa DK wa

w w a

(3.4d)



Eq. (3.4a), Eq. (3.4b), Eq. (3.4c) and Eq. (3.4d) will yield a non – trivial solution if the

determinant whose elements are the coefficient of these unknown constants is set equal to

zero. Thus we have

1 2 3 4

1 2 3 40

1 2 3 4

1 2 3 4

A A A A

B B B B

C C C C

D D D D

(3.5)

where,

21 ( ) sin cos

2 '( ) cos

3 0

4 0

A J uau a

jA J ua

u

A

A

(3.6a)

2

1 0

2 0

3 ( ) sin cos

4 '( ) cos

B

B

B K waw a

jB K wa

w

(3.6b)

2

2

1 ( ) cos sin

2 '( ) sin

3 ( ) cos sin

4 '( ) sin

C J uau a

jC J ua

u

C K waw a

jC K wa

w

(3.6c)



1

2

2

2

1 '( ) cos

2 ( ) sin cos

3 '( ) cos

4 ( ) sin cos

jD J ua

u

D J uau a

jD K wa

w

D K waw a

(3.6d)

Evaluation of the above determinant yields the following eigenvalue equation for β. The

determinant can be solve as

1 2 3 42 3 4 1 3 4 1 2 4 1 2 3

1 2 3 41 2 3 4 2 1 3 4 3 1 2 4 4 1 32 3

1 2 3 42 3 4 1 3 4 1 2 4 1 2 3

1 2 3 4

A A A AB B B B B B B B B B B B

B B B BA C C C A C C C A C C C A C C C

C C C CD D D D D D D D D D D D

D D D D

(3.7)

Using Eq. 3.6, we get

1 2 3 4 1 2 0 0

1 2 3 4 0 0 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

A A A A A A

B B B B B B

C C C C C C C C

D D D D D D D D

1 2 3 40 3 4 0 3 4

1 2 3 41 2 3 4 2 1 3 4

1 2 3 42 3 4 1 3 4

1 2 3 4

A A A AB B B B

B B B BA C C C A C C C

C C C CD D D D D D

D D D D

1 2 3 4

1 2 3 41 3 2 4 4 2 4( 2 3 3 2)

1 2 3 4

1 2 3 4

2 3 1 4 4 1 4( 1 3 3 1)

A A A A

B B B BA B C D C D B C D C D

C C C C

D D D D

A B C D C D B C D C D



1 2 3 4

1 2 3 41 3 2 4 1 3 4 2 1 4 2 3 1 4 3 2

1 2 3 4

1 2 3 4

2 3 1 4 2 3 4 1 2 4 1 3 2 4 3 1

A A A A

B B B BA B C D A B C D A B C D A B C D

C C C C

D D D D

A B C D A B C D A B C D A B C D

(3.8)

After eliminating unknown constants from Eq. (3.8) and Eq. (3.6), we get the following

characteristic equation.

1

2

2 2

2

2

2 2

2

2

( ) '( )sin cos cos

'( ) ( )

( ) '( )sin cos cos 0

'( ) ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(3.8)

Eq. (3.9) is standard characteristic equation, and is used for model dispersion properties

and model cutoff conditions.



Chapter 4

Results & Discussion

It is now possible to interpret the characteristic equation (Eq. 4.1) in numerical terms.

This will give us an insight into model properties of our waveguide.

1

2

2 2

2

2

2 2

2

2

( ) '( )sin cos cos

'( ) ( )


'( ) ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(4.1)

2 2 2

2

2 2

1 2

( / )k nawb

V n n

(4.2)

2 2

2 2 2 2 2 2 2

1 2

2 2( ) ( )

a aV u w a n n NA

(4.3)

where b & V are known as normalization propagation constant & normalized frequency

parameter respectively. We make some simple calculations based on Eq. 4.2 and Eq. 4.3.

We choose n1=1.50, n2=1.46 and λ =1.55µm. We take 1 for simplicity, but the result is

valid for any value of .

4.1 Dispersion characteristics

In order to plot the dispersion relations, we plot the normalized frequency parameter V

against the normalization propagation constant b. we considered five special cases

corresponding to the values of pitch angle ψ as 00, 30

0, 45

0, 60

0 and 90

0.




To obtain the dispersion curve for this case put ψ = 00 in Eq. 4.1. We now get

1 2

2 22 2

1 1 1 1

2 2

1 1 1 1

( ) '( ) ( ) '( )0

'( ) ( ) '( ) ( )

k kJ ua J ua K wa K wau w

J ua u a u J ua K wa w a w K wa

(4.4)

Dispersion curve corresponding to Eq. 4.4 is shown in Fig. 4.1.

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b

Fig. 4.1 Dispersion Curve for pitch angle ψ = 00





1

2

2 2

1 1

2

1 1

2 2

1 1

2

1 1

( ) '( )1 3 3

'( ) 2 2 4 ( )

( ) '( )1 3 30

'( ) 2 2 4 ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(4.5)


0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b






1

2

2 2

1 1

2

1 1

2 2

1 1

2

1 1

( ) '( )1 1 1

'( ) 2 ( )2 2

( ) '( )1 1 10

'( ) 2 ( )2 2

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(4.6)


0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b






1

2

2 2

1 1

2

1 1

2 2

1 1

2

1 1

( ) '( )3 1 1

'( ) 2 2 4 ( )

( ) '( )3 1 10

'( ) 2 2 4 ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(4.7)


0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b






1 2

2 2

1 1 1 1

1 1 1 1

( ) '( ) ( ) '( )0

'( ) ( ) '( ) ( )

k kJ ua J ua K wa K wau w

J ua u J ua K wa w K wa (4.8)


0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


From the above figures we observe that, they all have standard expected shape, but except

for lower order modes they comes in pairs, that is cutoff values for two adjacent mode

converge. This means that one effect of conducting helical winding is to split the modes

and remove a degeneracy which is hidden in conventional waveguide without windings.

We also observe that another effect of the conducting helical winding is to reduce the

cutoff values, thus increasing the number of modes. This effect is undesirable for the

possible use of these waveguide for long distance communication.



An anomalous feature in the dispersion curves is observable for ψ = 300, 45

0 and 60

0 for

this type of waveguide (Fig. 4.2, Fig. 4.3 and Fig. 4.4) near the lowest order mode. It is

found that on the left of the lowest cutoff values, portions of curves appear which have no

resemblance with standard dispersion curves, and have no cutoff values. This means that

for very small value of V anomalous dispersion properties may occur in helically wound

waveguides.

We found that some curves have band gaps of discontinuities between some value of V.

These represent the band gaps or forbidden bands of the structure. These are induced by

the periodicity of the helical windings.

4.2 Dependence of cutoff values Vc

We now come to table 4.1. we note particularly that the dependence of the cutoff V –

value (Vc) on ψ is such that as ψ is increased there is a drastic fall in Vc at ψ =300 and then

a small increase as ψ goes from 300 to 60

0; then is a quick rise as ψ changes from 60

0 to

900 (Fig. 4.6).

Table 4.1 Cutoff Vc values for some lower – order modes

ψ Vc Vc Vc Vc Vc Vc Vc Vc Vc

00 1.80 3.80 4.00 6.90 7.10 10.10 10.30 - -

300 0.05 1.70 1.80 3.70 3.90 7.00 7.10 10.20 10.30

450 0.40 1.70 1.80 3.65 3.70 7.00 7.20 10.20 10.30

600 0.30 1.50 1.80 3.70 3.90 7.00 7.20 10.20 10.30

900 1.90 3.80 5.40 7.00 8.60 10.20 11.80 - -



0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

Angle in Degree

Vc

Fig. 4.6 Dependence of cutoff values Vc on the pitch angle ψ

Thus the two most sensitive regions in respect of the influence of helical pitch angle ψ on

the cutoff values and the model properties of waveguides are ranges from ψ = 00 to ψ =

300 and ψ = 60

0 to ψ = 90

0 and these ranges of pitch angle expected to have potential

applications with ψ as a means for controlling the model properties.



Chapter 5

Conclusion & Future Work

5.1 Conclusion

From the above results (Fig. 4.1, Fig. 4.2, Fig. 4.3, Fig. 4.4 and Fig. 4.5) we observe that,

they all have standard expected shape, but except for lower order modes they comes in

pairs, that is cutoff values for two adjacent mode converge. This means that one effect of

conducting helical winding is to split the modes and remove a degeneracy which is

hidden in conventional waveguide without windings.

We also observe that another effect of the conducting helical winding is to reduce the

cutoff values, thus increasing the number of modes. This effect is undesirable for the

possible use of these waveguide for long distance communication.


0 and 60

0 for

this type of waveguide (Fig. 4.2, Fig. 4.3 and Fig. 4.4) near the lowest order mode. It is

found that on the left of the lowest cutoff values, portions of curves appear which have no

resemblance with standard dispersion curves, and have no cutoff values. This means that

for very small value of V anomalous dispersion properties may occur in helically wound

waveguides.

We found that some curves have band gaps of discontinuities between some value of V.

These represent the band gaps or forbidden bands of the structure. These are induced by

the periodicity of the helical windings.

Thus helical pitch angle controls the modal properties of this type of optical waveguide.

5.2 Future Work

In present work right handed helical winding is applied, left handed helical winding can

be applied and the effects on dispersion characteristics can be studied. Also in addition

left handed and right handed helical winding can be applied simultaneously to the fiber,



and result can be compared with present work. We have not studied polarization in this

work; this implies that the helical winding possible introduces important changes in

polarization properties. This can be considered for future work.

Present work consists the analysis and description of model characteristics,

considering 1 , for simplicity, although the result can be analyzed for any value of ,

so in future work more values of can be consider for more results. We can also consider

other type of fiber waveguides like, elliptical, triangular and square and study the model

characteristics for the mentioned waveguides and results can be compare.

Optical waveguides have their importance in versatile applications, viz. communication

purposes, sensing technology as well as integrated optical devices, so this type of

waveguides can be used for the above applications, this will surely improve the efficiency

and operation of the applied area.



References

[1] Jia Ming-Liu, “Photonic Devices”, Cambridge University Press, UK, 2005.

[2] http://www.bb-elec.com/tech_articles/fiber_optic_technology.asp

[3] Kumar, D. and O. N. Singh II, “Towards the dispersion relations for

dielectric optical fibers with helical windings under slow and fast wave

considerations – a comparative analysis,” PIER, Vol. 80, 409–420, 2008.

[4] Kumar, D. and O. N. Singh II, “An analytical study of the modal

characteristics of annular step – index fiber of elliptical cross – section with

two conducting helical windings on the two boundary surfaces between the

guiding and non – guiding regions” Optik, Vol. 113, No. 5, 193-196, 2002.

[5] Singh, U. N., O. N. Singh II, P. Khastgir and K. K. Dey “Dispersion

characteristics of helically cladded step – index optical fiber analytical

study” J. Opt. Soc. Am. B, 1273-1278, 1995.

[6] M. P. S. Rao, Vivek Singh, B. Presad and S. P. Ojha “Model characteristic

and dispersion curves of hypocycloidal optical waveguide” Optik, 110, No.

2, 81-85, 1999.

[7] Ajoy Ghatak and K. Thyagarajan, “Optical Electronics” Cambridge

University Press, India, 2008.

[8] http://commons.wikimedia.org/wiki/File:Fiber_optic_numerical_aperture.s

vg

[9] http://www.its.bldrdoc.gov/projects/devglossary/alldef2.html

[10] http://www.daviddarling.infochildrens_encyclopedialight_Chapter2.html

[11] http://www.rp-photonics.com/dispersion_compensation.html

[12] http://www.fiberoptics4sale.com/wordpress/fiber-dispersion-and-optical-

dispersion-an-overview/

[13] http://www.tpub.com/neets/tm/106-10.htm

[14] Keiser G.,“Optical Fiber Communications”, Chap.2, 3rd

edition McGraw-

Hill, Singapore, 2000.

[15] http://en.wikipedia.org/wiki/Bessel_function

[16] http://www.cdeep.iitb.ac.in/nptel/Electrical%20&%20Comm%20Engg/Opt

ical%20Communication/Course_home-M3.html



[17] Kumar, D. and O. N. Singh II, “Modal characteristic equation and

dispersion curves for an elliptical step – index fiber with a conducting

helical winding on the core – cladding boundary – An analytical study ”

IEEE, Journal of Light Wave Technology, Vol. 20, No.8, 1416-1424, USA,

August 2002.

[18] http://en.wikipedia.org/wiki/Helix

[19] http://www.kadee.com/htmbord/page636.htm

[20] http://commons.wikimedia.org/wiki/File:Helix_diagram.png

[21] http://en.wikipedia.org/wiki/Corkscrew_(Cedar_Point)

[22] http://en.wikipedia.org/wiki/Torsion_of_a_curve

[23] Kumar, D. and O. N. Singh II, “Some special cases of propagation

characteristics of an elliptical step – index fiber with a conducting helical

winding on the core – cladding boundary – An analytical treatment ,” Optik

Vol. 112, No. 12, 561-566, 2001.

[24]

[25]

http://en.wikipedia.org/wiki/Cylindrical_coordinate_system

Govind P. Agrawal, “Fiber – Optic Communication Systems”, 3rd

edition A

John Wiley & Sons, Inc., Publication, New York, 2002.



Publication

[1]

Ajay Kumar Gautam, Dr. Vivekanand Mishra and Prof. B.R. Taunk, “Dispersion

Characteristic of Optical Waveguide with Helical Winding for Different Pitch

Angle”, National Conference on Electronics, Communication & Instrumentation,

CSE Jhansi, e – Manthan, 2010, 71 – 73, 2nd

- 3rd

April 2010.

Publication (Under Communication)

[2]

Ajay Kumar Gautam, Dr. Vivekanand Mishra and Prof. B.R. Taunk, “Modal

Dispersion Characteristics of Circular Optical Waveguide with Helical Winding -

A Comparison for Different Pitch Angles”, International Conference on Advances

in Computing and Communication, NIT Hamirpur, 2010.

[3] V. Mishra, A. K. Gautam, B. R. Taunk, “Effect of Helical Pitch Angles on

Dispersion Characteristics of Circular Optical Waveguide having Helical Windings

on Core - Cladding Interface”, International Journal for Laser Physics, 2010.

[4] V. Mishra, A. K. Gautam, B. R. Taunk, “Dispersion & Cuttoff Characteristics of

Circular Helically Cladded Optical Fiber” International Journal on Engineering &

Technology, 2010.

i

ii

iii

iv

v

vi

vii

viii

ix

x

xi

xii

xiii

xiv

xv

xvi

xvii

DISPERSSION CHARACTERISTIC OF OPTICAL

WAVEGUIDE WITH HELICAL WINDING FOR DIFFERENT

PITCH ANGLE

Ajay Kumar Gautam

ECED, SVNIT

Surat , India

[email protected]

Dr. Vivekanand Mishra

ECED, SVNIT

Surat , India

[email protected]

Prof. B. R. Taunk

ECED, SVNIT

Surat , India

[email protected]

Abstract- The model dispersion characteristics of circular optical waveguide with helical

winding at core-cladding interface are obtained for different pitch angle. This paper gives

the idea to obtain dispersion characteristics, and compression of dispersion characteristics

at different pitch angles. We obtained the dispersion characteristics by using boundary

condition and this condition have been utilized to get the model Eigen values equation.

From these Eigen value equations dispersion curve are obtained and plotted for two

particular values of the pitch angle of the winding and the result has been compared.

Keywords- Optical fiber communication, optical fiber dispersion, helical winding, helix

pitch angle.

I. Introduction

Optical fibers with helical winding are known as complex optical waveguides. The use of

helical winding in optical fibers makes the analysis much accurate [1]. As the number of

propagating modes depends on the helix pitch angle [2], so helical winding at core-cladding

interface can control the dispersion characteristics [3-7] of the optical waveguide. The

winding angle of helix (ψ) can take any arbitrary value between 0 to π/2. In case of sheath

helix winding [1], cylindrical surface with high conductivity in the direction of winding

which winds helically at constant pitch angle (ψ) around the core cladding boundary surface.

We assume that the waveguide have real constant refractive index of core and cladding is n1

and n2 respectively (n1 > n2). In this type of optical wave guide which we get after winding,

the pitch angle controls the model characteristics of optical waveguide.

2. Theoretical Analysis

We can take a case of a fiber with circular cross-section wound with a sheath helix at the

core-clad interface (Fig. 1). A sheath helix can be assumed by winding a very thin conducting

wire around the cylindrical surface so that the spacing between the nearest two windings is

very small and yet they are insulated from each another. In our structure, the helical windings

are made at a constant helix pitch angle (ψ). We assume that (n1-n2) / n1 << 1.

xviii

Fig. 1 Circular Optical Waveguide with conducting helical winding at core cladding

interface

3. Boundary Conditions

The tangential component of the electric field in the direction of winding should be zero, and

tangential component of both the electric field and magnetic field in the direction

perpendicular to the winding must be continuous. So we consider the following boundary

conditions [8].

Ez1 sin ψ + Eϕ1 cos ψ = 0 (1)

Ez2 sin ψ + Eϕ2 cos ψ = 0 (2)

(Ez1 - Ez2) cos ψ - (Eϕ1 - Eϕ2) sin ψ = 0 (3)

(Hz1 - Hz2) sin ψ + (Hϕ1 - Hϕ2) cos ψ= 0 (4)

4. Circular Optical Waveguide with conducting helical Winding

The guided modes with this type of fiber can be analyzed in cylindrical coordinate system (r,

ϕ, z). Where z is the direction of wave propagation i.e. along the axis of the optical fiber. The

most important condition to have guided field is, n2k < β < n1k and must be satisfied, where

n1 and n2 are the refractive indices of the core and cladding region respectively and k is free

space propagation constant (k = 2π/λ, k2 = n2k and k1 = n1k) . In core region we take the

solution of linear combination of Bessel function of first kind {Jν (x)}, whereas for cladding

region we take modified Bessel function of second kind {Kν (x)} [9]. We take ν = 1, for

lower order guided mode index. The axial field components for core region can be can

written as

Ez1 = AJ1 (Ua) F(ϕ) e j (ωt−βz)

(5)

Hz1 = BJ1 (Ua) F(ϕ) e j(ωt−βz)

(6)

The axial field components for clad region can be can written as

Ez2 = CK1 (Wa) F(ϕ)e j (ωt−βz)

(7)

Hz2 = DK1 (Wa) F(ϕ)e j (ωt−βz)

(8)

xix

Also

U2 = k1

2 – β

2 =

21 - β

2 (9)

W2 = β

2 – k2

2 = β

2 -

22 (10)

Where β is the axial component of propagation vector, is the wave frequency, is the

permeability of the non-magnetic medium, 1 and 2 are the permittivity of the core and

cladding region respectively, and A, B, C and D are unknown constant and will be

determinant, F(ϕ) is the function of coordinate ϕ,

Now use Maxwell`s equation to obtain transverse components of the electric field and

magnetic field. So transverse components of the electric field and magnetic field Eϕ1 and Hϕ1

for core region can be written as

Eϕ1 = - ( j/U2

) [ j(β/a) AJ1 ( Ua ) - UBJ`1 ( Ua ) ] F(ϕ)e j (ωt−βz)

(11)

Hϕ1 = - ( j/U2

) [ j(β/a) BJ1 ( Ua ) + 1UAJ`1 ( Ua ) ] F(ϕ)e j (ωt−βz)

(12)

And transverse components of the electric field and magnetic field Eϕ1 and Hϕ1 for cladding

region can be written as

Eϕ2 = - ( j/W2

) [ j(β/a) CK1 ( Wa ) - WDK`1 ( Wa ) ] F(ϕ)e j (ωt−βz)

(13)

Hϕ2 = - ( j/W2

) [ j(β/a) DJ1 ( Wa ) + 2WCK`1 ( Wa ) ] F(ϕ)e j (ωt−βz)

(14)

Now eliminate the field components Eϕ1, Hϕ1, Eϕ2, and Hϕ2 from boundary conditions (1) to

(4) and field component equations (11) to (14). We get four equations which involves four

unknown constants A, B, C and D. Now we put coefficient of these unknown constants A, B,

C and D into determinant to solve these four equations.

Now put Δ = 0. This will produce non-trivial solution.

Δ=0 (13)

Where A1 to A4 coefficients of A, B1 to B4 coefficients of B, C1 to C4 coefficients of D and

D1 to D4 coefficients of D.

After simplifying the determinant, we get a simplified equation for lowest order modes.

U [ J1 ( Ua ) / J1`( Ua ) ] [ sin ψ + (β cos ψ ) / U2a]

2

- W [ K1 ( Wa ) / K1`( Wa ) ] [ sin ψ + (β cos ψ ) / U2a]

2

- [ k12 / U] [ J`1 ( Ua ) / J1 ( Ua ) ] cos

2 ψ

+ [ k22 / U] [ K`1 ( Ua ) / K1 ( Ua ) ] cos

2 ψ = 0 (14)

A1 B1 C1 D1

A2 B2 C2 D2

A3 B3 C3 D3

A4 B4 C4 D4

xx

We use equation (14) to plot dispersion characteristics of an optical waveguide with helical

winding. We can plot dispersion characteristics for different pitch angles (ψ). We take two

values of ψ. However the equation is valid for any value of pitch angle (ψ). We can also use

equation (14) to find the value of β.

5. Result

We plot the dispersion characteristics (V versus β/k) of the waveguide with helical winding.

So we find the different value of β by using equation (14) for two different pitch angles (ψ),

i.e. 0 and π/2. Here V is called normalized frequency [10] (also V parameter or V number), it

is dimensionless numbers, as it determines how many modes a fiber can support. V is given

by

V2 = (2πa / λ)

2 (n1

2 – n2

2) (15)

Relation between β/k and V is given by normalized propagation constant (b), and is given by,

b = [(β/k)2 – n2

2] / [n1

2 – n2

2] (16)

Now we plot the dispersion characteristics for two different values of pitch angle as shown in

figure 2 and 3. For this we use n1 = 1.5, n2 = 1.46, and the λ = 1.55 µm. First we consider

helical pitch angle ψ = 00. It means winding is perpendicular to the axis of the fiber, we can

see obtained cutoff values for some modes as shown in dispersion curve (Fig. 2).

Fig. 2 Dispersion curve for ψ = 00 Fig. 3 Dispersion curve for ψ = 90

0

Secondly we consider helical pitch angle ψ = 900. It means winding is parallel to the axis of

the fiber, we can see obtained cutoff values for some modes as shown in dispersion curve

(Fig. 3). We observed that these two curves have different cutoff values. We observed that

the cutoff value for helical pitch angle ψ = 900 is somewhat higher than that for helical pitch

angle ψ = 00.

6. Conclusion

xxi

From these two results we observed that on increasing the helical pitch angle (ψ), we get

shifting of model cutoff to higher value. Which means on increasing the pitch angle, we get

reduction in the number of sustained modes.

References

[1] Kumar, D. and O. N. Singh II, “Modal characteristics equation and dispersion curves

for an elliptical step-index fiber with a conducting helical winding on the

core-cladding boundary - An analytical study,” IEEE, Journal of Light Wave

Technology, Vol. 20, No. 8, 1416–1424, USA, August 2002.

[2] Watkins, D. A., Topics in Electromagnetic Theory, John Wiley and Sons Inc., NY, 1958.

[3] V.N. Mishra, Vivek Singh, B. Prasad, S. P. Ojha, “Optical Dispersion curves of two metal

- clad lightguides having double convex lens core cross sections”, Wiley, Microwave and

Optical Technology Letters, Vol. 24, No. 4, 229-232, New York, Feb 20, 2000.

[4] V.N. Mishra, V. Singh, B. Prasad, S. P. Ojha, “An Analytical investigation of dispersion

characteristic of a lightguide with an annular core cross section bounded by two

cardioids”, Wiley, Microwave and Optical Technology Letters, Vol. 24, No. 4, 229-232,

New York, Feb 20, 2000.

[5] V. Singh, S. P. Ojha, B. Prasad, and L. K. Singh, “Optical and microwave Dispersion

curves of an optical waveguide with a guiding region having a core cross section with a

lunar shape”, Optik 110, 267-270, 1999.

[6] V. Singh, S. P. Ojha, and L. K. Singh, “Model Behaviour, cutoff condition, and

dispersion characteristics of an optical waveguide with a core cross section bounded by

two spirals”, microwave Optical Technology Letter, Vol. 21, 121-124, 1999.

[7] V. Singh, S. P. Ojha, and B. Prasad, “weak guidance modal dispersion characteristics of

an optical waveguide having core with sinusoidally varying gear shaped cross section”,

microwave Optical Technology Letter, Vol. 22, 129-133, 1999.

[8] Gloge D., “Dispersion in weakly guiding fibers,” Appl. Opt., Vol. 10, 2442 - 2445,

1971.

[9] P. K. Choudhury, D. Kumar, and Z. Yusoff, F. A. Rahman, “An analytical investigation

of four-layer dielectric optical fibers with au nano-coating - A comparison with three-

layer optical fibers”, PIER 90, 269 - 286, 2009.


edition McGraw-Hill,

Singapore, 2000.

xxii

Modal Dispersion Characteristics

of Circular Optical waveguide

with helical winding - A Comparison,

for different pitch angles Ajay Kumar Gautam

#, Dr. Vivekanand Mishra

*, B. R. Taunk

#

*,#Electronics Engineering Department, SVNIT

Surat India #[email protected]

*[email protected]

#[email protected]

*Sr. Member IEEE

Abstract— In this paper dispersion characteristic of

conventional optical waveguide with helical winding at core –

cladding interface has been obtained. The model dispersion

characteristics of optical waveguide with helical winding at core-

cladding interface have been obtained for five different pitch

angles. This paper includes dispersion characteristics of optical

waveguide with helical winding, and compression of dispersion

characteristics of optical waveguide with helical winding at core-

cladding interface for five different pitch angles. Boundary

conditions have been used to obtain the dispersion characteristics

and these conditions have been utilized to get the model Eigen

values equation. From these Eigen value equations dispersion

curve are obtained and plotted for modified optical waveguide

for particular values of the pitch angle of the winding and the

result has been compared.

Keywords — Optical fiber communication, fiber dispersion,

helical winding, helix pitch angle, modal cut-off I. INTRODUCTION

Optical fibers with helical winding are known as complex

optical waveguides. The conventional optical fiber having a

circular core cross – section which is widely used in optical

communication systems. The use of helical winding in optical

fibers makes the analysis much accurate [1]. The propagation

characteristics of optical fibers with elliptic cross – section

have been investigated by many researchers. Singh [13] have

proposed an analytical study of dispersion characteristics of

helically cladded step – index optical fiber with elliptical core.

Present work is the study of circular optical waveguide with

sheath helix in between the core and cladding region, this

work also gives the comparison of dispersion characteristic at

different pitch angles. The sheath helix [12] is a cylindrical

surface with high conductivity in a preferential direction

which winds helically at constant angle around the core –

cladding boundary surfaces. As the number of propagating

modes depends on the helix pitch angle [2], so helical winding

at core-cladding interface can control the dispersion

characteristics [3-7] of the optical waveguide. The winding

angle of helix (ψ) can take any arbitrary value between 0 to

π/2. In case of sheath helix winding [1], cylindrical surface

with high conductivity in the direction of winding which

winds helically at constant pitch angle (ψ) around the core

cladding boundary surface. We assume that the waveguide

have real constant refractive index of core and cladding is n1

and n2 respectively (n1 > n2). In this type of optical wave

guide which we get after winding, the pitch angle controls the

model characteristics of optical waveguide.

II. THEORETICAL ANALYSIS

The optical waveguide is the fundamental element that

interconnects the various devices of an optical integrated

circuit, just as a metallic strip does in an electrical integrated

circuit. However, unlike electrical current that flows through a

metal strip according to Ohm’s law, optical waves travel in

the waveguide in distinct optical modes. 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. The mode theory [10] is used to

describe the properties of light that ray theory is unable to

explain. The mode theory uses electromagnetic wave behavior

to describe the propagation of light along a fiber. A set of

guided electromagnetic waves is called the modes [13, 16] of

the fiber. For a given mode, a change in wavelength can

prevent the mode from propagating along the fiber. The mode

xxiii

is no longer bound to the fiber. The mode is said to be cut off

[13]. Modes that are bound at one wavelength may not exist at

longer wavelengths. The wavelength at which a mode ceases

to be bound is called the cutoff wavelength [11] for that

mode. However, an optical fiber is always able to propagate at

least one mode. This mode is referred to as the fundamental

mode [16] of the fiber. The fundamental mode can never be

cut off. We can take a case of a fiber with circular cross-

section wound with a sheath helix at the core-clad interface

(Fig. 1). A sheath helix can be assumed by winding a very thin

conducting wire around the cylindrical surface so that the

spacing between the nearest two windings is very small and

yet they are insulated from each another. In our structure, the

helical windings are made at a constant helix pitch angle (ψ).

We assume that (n1-n2) / n1 << 1.

III. BOUNDARY CONDITIONS

The tangential component of the electric field in the

direction of winding should be zero, and tangential component

of both the electric field and magnetic field in the direction

perpendicular to the winding must be continuous. So we

consider the following boundary conditions [8].

1 1 0zE sin E cos (1)


1 2 1 2 0z zE E cos E E sin (3)

1 2 1 2 0z zH H sin H H cos (4)

IV. FIBER WITH HELICAL WINDING

The guided modes with this type of fiber can be analysed in

cylindrical coordinate system (r, ϕ, z), where z is the direction

of wave propagation i.e. along the axis of the optical fiber.

The most important condition to have guided field is, n2k < β

< n1k and must be satisfied, where n1 and n2 are the refractive

indices of the core and cladding region respectively and k is

free space propagation constant (k = 2π/λ, k2 = n2k and k1 =

n1k) . In core region we take the solution of linear combination

of Bessel function of first kind {Jν (x)}, whereas for cladding

region we take modified Bessel function of second kind {Kν

(x)} [9]. We take ν = 1, for lower order guided mode index.

The axial field components for core region can be can written

as,

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

a)

( )

1 1

j t z

zE AJ Ua F e (5)

( )

1 1 j t z

zH BJ Ua F e (6)

The expressions for Ez and Hz outside the core are, when

(r > a),

( )

2 1

j t z

zE CK Wa F e (7)

( )

2 1

j t z

zH DK Wa F e (8)

Where,

2 2 2 2 2

1 1U –k (9)

2 2 2 2 2

2 2W k (10)

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

evaluated from the boundary conditions. Also J ua

and ( )K wa are the Bessel functions. For a guided mode, the

propagation constant lies between two limits 2 and

1 . If

2 2 1 1n k k k n k then a field distribution is generated

which will has an oscillatory behaviour in the core and a

decaying behaviour in the cladding. The energy then is

propagated along fiber without any loss. Where 2

k

is

free – space propagation constant. The transverse field

components can be obtained by using Maxwell’s standard

relations. So the electric and magnetic field components Eϕ

and Hϕ can be written as,

Fig. 1 Fiber with conducting helical winding at core cladding interface

Where β is the axial component of propagation vector, is

the wave frequency, is the permeability of the non-magnetic

medium, 1 and 2 are the permittivity of the core and cladding

region respectively, and A, B, C and D are unknown constant

and will be determinant, F(ϕ) is the function of coordinate ϕ,

Now use Maxwell`s equation [10, 19] to obtain transverse

components of the electric field and magnetic field, so

transverse components of the electric field and magnetic field

Eϕ1 and Hϕ1 for core region can be written as.

2 ( )

1

'

1 1

/

/

j t zE j U F e

j a AJ Ua UBJ Ua

(11)

2 ( )

1

'

1 1 1

/

/

j t zH j U F e

j a BJ Ua UAJ Ua

(12)

The axial field components of the electric field and

magnetic field Eϕ2 and Hϕ2 for clad region can be can written

as

2 ( )

2

'

1 1

/

/

j t zE j W F e

j a CK Wa WDK Wa

(13)

2 ( )

2

'

1 2 1

/

/

j t zH j U F e

j a DK Wa WCK Wa

(14)

xxiv

Now eliminate the field components Eϕ1, Hϕ1, Eϕ2, and Hϕ2

from boundary conditions (1) to (4) and field component

equations (11) to (14). We get four equations which involves

four unknown constants A, B, C and D.

2( ) sin cos '( ) cos 0

jAJ ua BJ ua

u a u

(15)

2( ) sin cos '( ) cos 0

jCK wa DK wa

w a w

(16)

2

2



jAJ ua BJ ua

u a u

jCK wa DK wa

w a w

(17)

1

2

2

2



jAJ ua BJ ua

u u a

jCK wa DK wa

w w a

(18)

Eq. (15), Eq. (16), Eq. (17) and Eq. (18) will yield a non –

trivial solution if the determinant whose elements are the

coefficient of these unknown constants is set equal to zero.

Thus we have

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

A A A A

B B B B

C C C C

D D D D

(19)

where,

21 ( ) sin cos

2 '( ) cos

3 0

4 0

A J uau a

jA J ua

u

A

A

(20)

2

1 0

2 0

3 ( ) sin cos

4 '( ) cos

B

B

B K waw a

jB K wa

w

(21)

2

2

1 ( ) cos sin

2 '( ) sin

3 ( ) cos sin

4 '( ) sin

C J uau a

jC J ua

u

C K waw a

jC K wa

w

(22)

1

2

2

2

1 '( ) cos

2 ( ) sin cos

3 '( ) cos

4 ( ) sin cos

jD J ua

u

D J uau a

jD K wa

w

D K waw a

(23)

After simplifying the determinant, we get a simplified

equation for lowest order modes.

1

2

2 2 '21 1

' 2

1 1

2 2 '21 1

' 2

1 1

( ) ( )sin cos cos

( ) ( )

( ) ( )sin cos cos 0

( ) ( )

kJ Ua J UaU

J Ua U a U J Ua

kK Wa K WaW

K Wa W a W K Wa

(20)

We use equation (20) to plot dispersion characteristics [12-

18] of an optical waveguide with helical winding. We can plot

dispersion characteristics for different pitch angles (ψ). We

take five different values of ψ. However the equation is valid

for any value of pitch angle (ψ).

V. RESULTS

It is now possible to interpret the characteristic equation

(Eq. 20) in numerical terms. This will give us an insight into

model properties of our waveguide. For this we can use

following relations,

1

2

2 2 '21 1

' 2

1 1

2 2 '21 1

' 2

1 1

( ) ( )sin cos cos

( ) ( )

( ) ( )sin cos cos 0

( ) ( )

kJ Ua J UaU

J Ua U a U J Ua

kK Wa K WaW

K Wa W a W K Wa

(21)

2 2 2

2

2 2

1 2

( / )k nawb

V n n

(22)

2

2 2 2 2 2 2

1 2

2( ) ( )

aV u w a n n

(23)

where b & V are known as normalization propagation

constant & normalized frequency parameter respectively. We

make some simple calculations based. We choose n1=1.50,

n2=1.46 and λ =1.55µm.

A. Dispersion Curve

1. Dispersion Curve for for pitch angle ψ = 00

xxv

1

2

2 2

1 1

2

1 1

2 2

1 1

2

1 1

( ) '( )

'( ) ( )

( ) '( )0

'( ) ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b

Fig. 2 Dispersion Curve for pitch angle ψ = 00


1

2

2 2

1 1

2

1 1

2 2

1 1

2

1 1

( ) '( )1 3 3

'( ) 2 2 4 ( )

( ) '( )1 3 30

'( ) 2 2 4 ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b



1

2

2 2

1 1

2

1 1

2 2

1 1

2

1 1

( ) '( )1 1 1

'( ) 2 ( )2 2

( ) '( )1 1 10

'( ) 2 ( )2 2

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b



1

2

2 2

1 1

2

1 1

2 2

1 1

2

1 1

( ) '( )3 1 1

'( ) 2 2 4 ( )

( ) '( )3 1 10

'( ) 2 2 4 ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b



1

2

2

1 1

1 1

2

1 1

1 1

( ) '( )

'( ) ( )

( ) '( )0

'( ) ( )

kJ ua J uau

J ua u J ua

kK wa K waw

K wa w K wa

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


From the above figures we observe that, they all have

standard expected shape, but except for lower order modes

they comes This effect is undesirable for the possible use of.

This means that one effect of conducting helical winding is to

these waveguide for long distance communication in pairs that

is cut-off values for two adjacent modes converge.

xxvi

An anomalous feature in the dispersion curves is

observable for ψ = 300, 45

0 and 60

0 for this type of waveguide

(Fig. 3, Fig. 4 and Fig. 5) near the lowest order mode. It is

found that on the left of the lowest cut-off values, portions of

curves appear which have no resemblance with standard

dispersion curves, and have no cut-off values. This means that

for very small value of V anomalous dispersion properties

may occur in helically wound waveguides.

We found that some curves have band gaps of

discontinuities between some value of V. These represent the

band gaps or forbidden bands of the structure. These are

induced by the periodicity of the helical windings.

B. Dependence of Cut-off values on Pitch Angle

We now come to Table I we note particularly that the

dependence of the cut-off V – value (Vc) on ψ is such that as ψ

is increased there is a drastic fall in Vc at ψ =300 and then a

small increase as ψ goes from 300 to 60

0; then is a quick rise

as ψ changes from 600 to 90

0 (Fig. 7).

TABLE I

CUT-OFF Vc VELUES FOR SOME LOWER – ORDER

MODES


00 1.80 3.80 4.00 6.90 7.10 10.10 10.30 - -

300 0.05 1.70 1.80 3.70 3.90 7.00 7.10 10.20 10.30

450 0.40 1.70 1.80 3.65 3.70 7.00 7.20 10.20 10.30

600 0.30 1.50 1.80 3.70 3.90 7.00 7.20 10.20 10.30

900 1.90 3.80 5.40 7.00 8.60 10.20 11.80 - -

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

Angle in Degree

Vc

Fig. 7 Dependence of cut-off values Vc on the pitch angle ψ

Thus the two most sensitive regions in respect of the influence

of helical pitch angle ψ on the cutoff values and the model

properties of waveguides are ranges from ψ = 00 to ψ = 30

0

and ψ = 600 to ψ = 90

0 and these ranges of pitch angle

expected to have potential applications with ψ as a means for

controlling the model properties split the modes and remove a

degeneracy which is hidden in conventional waveguide

without windings We also observe that another effect of the

conducting helical winding is to reduce the cutoff values, thus

increasing the number of modes.

VI. CONCLUSIONS

From the above results (Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6)

we observe that, they all have standard expected shape, but

except for lower order modes they comes in pairs, that is cut-

off values for two adjacent mode converge. This means that

one effect of conducting helical winding is to split the modes

and remove a degeneracy which is hidden in conventional

waveguide without windings.

We also observe that another effect of the conducting helical

winding is to reduce the cut-off values, thus increasing the

number of modes. This effect is undesirable for the possible

use of these waveguide for long distance communication.

We found that some curves have band gaps of discontinuities

between some value of V. These represent the band gaps or

forbidden bands of the structure. These are induced by the

periodicity of the helical windings. Thus helical pitch angle

controls the modal properties of this type of optical waveguide.

From the above discussions we can conclude that the

modal cut-off for helical pitch angle ψ = 300, 45

0 and 60

0 are

higher than the modal cut-off for helical pitch angle ψ = 00

and 900. This means, for some specific range of cut-off values

Vc, one can have greater number of modes for helical pitch

angle ψ = 300, 45

0 and 60

0 than for helical pitch angle ψ = 0

0

and 900. So the helical pitch angle ψ = 30

0, 45

0 and 60

0 are

better than helical pitch angle ψ = 00 and 90

0.

REFERENCES

[1] Kumar, D. and O. N. Singh II, ―Modal characteristics equation and

dispersion curves for an elliptical step-index fiber with a

conducting helical winding on the core-cladding boundary - An

analytical study,‖ IEEE, Journal of Light Wave Technology, Vol. 20,

No. 8, 1416–1424, USA, August 2002.

[2] Watkins, D. A., Topics in Electromagnetic Theory, John Wiley and

Sons Inc., NY, 1958.

[3] V.N. Mishra, Vivek Singh, B. Prasad, S. P. Ojha, ―Optical Dispersion

curves of two metal - clad lightguides having double convex lens core

cross sections‖, Wiley, Microwave and Optical Technology Letters, Vol.

24, No. 4, 229-232, New York, Feb 20, 2000.

[4] V.N. Mishra, V. Singh, B. Prasad, S. P. Ojha, ―An Analytical

investigation of dispersion characteristic of a lightguide with an

annular core cross section bounded by two cardioids‖, Wiley,

Microwave and Optical Technology Letters, Vol. 24, No. 4, 229-232,

New York, Feb 20, 2000.

[5] V. Singh, S. P. Ojha, B. Prasad, and L. K. Singh, ―Optical and

microwave Dispersion curves of an optical waveguide with a guiding

xxvii

region having a core cross section with a lunar shape‖, Optik 110, 267-

270, 1999.

[6] V. Singh, S. P. Ojha, and L. K. Singh, ―Model Behaviour, cutoff

condition, and dispersion characteristics of an optical waveguide with a

core cross section bounded by two spirals‖, microwave Optical

Technology Letter, Vol. 21, 121-124, 1999.

[7] V. Singh, S. P. Ojha, and B. Prasad, ―weak guidance modal dispersion

characteristics of an optical waveguide having core with sinusoidally

varying gear shaped cross section‖, microwave Optical Technology

Letter, Vol. 22, 129-133, 1999.

[8] Gloge D., ―Dispersion in weakly guiding fibers,‖ Appl. Optics, Vol.

10, 2442 - 2445, 1971.

[9] P. K. Choudhury, D. Kumar, and Z. Yusoff, F. A. Rahman, ―An

analytical investigation of four-layer dielectric optical fibers with au

nano-coating - A comparison with three-layer optical fibers‖, PIER 90,

269 - 286, 2009.

[10] Keiser G.,―Optical Fiber Communications‖, Chap.2, 3rd edition

McGraw-Hill,Singapore,2000.

[11] Jia Ming-Liu, ―Photonic Devices‖, Cambridge University Press, UK,

2005.

[12] Kumar, D. and O. N. Singh II, ―Towards the dispersion relations for

dielectric optical fibers with helical windings under slow and fast wave

considerations – a comparative analysis,‖ PIER, Vol. 80, 409–420,

2008.

[13] Kumar, D. and O. N. Singh II, ―An analytical study of the modal

characteristics of annular step – index fiber of elliptical cross – section

with two conducting helical windings on the two boundary surfaces

between the guiding and non – guiding regions‖ Optik, Vol. 113, No. 5,

193-196, 2002.

[14] Singh, U. N., O. N. Singh II, P. Khastgir and K. K. Dey ―Dispersion

characteristics of helically cladded step – index optical fiber analytical

study‖ J. Opt. Soc. Am. B, 1273-1278, 1995.

[15] M. P. S. Rao, Vivek Singh, B. Presad and S. P. Ojha ―Model

characteristic and dispersion curves of hypocycloidal optical

waveguide‖ Optik, 110, No. 2, 81-85, 1999.

[16] Ajoy Ghatak and K. Thyagarajan, ―Optical Electronics‖ Cambridge

University Press, India, 2008.

[17] Kumar, D. and O. N. Singh II, ―Some special cases of propagation

characteristics of an elliptical step – index fiber with a conducting

helical winding on the core – cladding boundary – An analytical

treatment ,‖ Optik Vol. 112, No. 12, 561-566, 2001.

[18] Kumar, D. and O. N. Singh II, ―Modal characteristic equation and

dispersion curves for an elliptical step – index fiber with a conducting

helical winding on the core – cladding boundary – An analytical

study ‖ IEEE, Journal of Light Wave Technology, Vol. 20, No.8, 1416-

1424, USA, August 2002.

[19] Govind P. Agrawal, ―Fiber – Optic Communication Systems‖, 3rd

edition, A John Wiley & Sons, Inc., Publication, New York, 2002.

xxviii

Effect of Helical Pitch Angles on Dispersion Characteristics of Circular

Optical Waveguide Having Helical Windings on Core - Cladding Interface

V. Mishraa,*

, A. K. Gautama, and B. R. Taunk

a

Electronics Engineering Department, Sardar Vallabhbhai National Institute of Technology Surat, India *Sr. Member IEEE

*Email: [email protected]

Abstract – This article includes dispersion characteristics of optical waveguide with helical

winding, and compression of dispersion characteristics of optical waveguide with helical

winding at core-cladding interface for five different pitch angles. In this article dispersion

characteristic of conventional optical waveguide with helical winding at core – cladding

interface has been obtained. The model dispersion characteristics of optical waveguide with

helical winding at core-cladding interface have been obtained for five different pitch angles.

Boundary conditions have been used to obtain the dispersion characteristics and these

conditions have been utilized to get the model Eigen values equation. From these Eigen value

equations dispersion curve are obtained and plotted for modified optical waveguide for

particular values of the pitch angle of the winding and the effect of this winding has been

discussed. The article also shows the effect in the Dispersion Curve with changing the Pitch

Angle.

Keywords – Bessel functions, dispersion curves, characteristics equation, sheath helix,

circular waveguide, modal cutoff.

TOPIC – Fiber Optics

xxix

1. INTRODUCTION

An optical waveguide is basically a cylindrical dielectric waveguide with a circular cross section where a high-

index wave guiding core is surrounded by a low-index cladding. The index step and profile are controlled by the

concentration and distribution of dopants. Silica fibers are ideal for light transmission in the visible and near-

infrared regions because of their low loss and low dispersion in these spectral regions. They are therefore

suitable for optical communications. Even though optical fiber seems quite flexible, it is made of glass, which

cannot withstand sharp bending or longitudinal stress. Therefore when fiber is placed inside complete cables

special construction techniques are employed to allow the fiber to move freely within a tube. Usually fiber optic

cables contain several fibers, a strong central strength member and one or more metal sheaths for mechanical

protection. Some cables also include copper pairs for auxiliary applications. Optical fibers with helical winding

are known as complex optical waveguides. The use of helical winding in optical fibers makes the analysis much

accurate. As the number of propagating modes depends on the helix pitch angle, so helical winding at core –

cladding interface can control the dispersion characteristics of the optical waveguide [3].

The conventional optical fiber having a circular core cross – section which is widely used in optical

communication systems [1]. Recently metal – clad optical waveguides have been studied because these provide

potential applications, connecting the optical components to other circuits. Metallic – cladding structure on an

optical waveguide is known as a TE – mode pass polarizer and is commercially applied to various optical

devices [4]. The propagation characteristics of optical fibers with elliptic cross – section have been investigated

by many researchers. Singh [5] have proposed an analytical study of dispersion characteristics of helically

cladded step – index optical fiber with circular core. The model characteristic and dispersion curves of a

hypocycloidal optical waveguide have been investigated by Ojha [6]. Present work is the study of circular

optical waveguide with sheath helix [3] in between the core and cladding region. The sheath helix is a

cylindrical surface with high conductivity in a preferential direction which winds helically at constant angle

around the core – cladding boundary surfaces.

Optical fibers with helical winding are known as complex optical waveguides. The conventional optical fiber

having a circular core cross – section which is widely used in optical communication systems. The use of helical

winding in optical fibers makes the analysis much accurate [1]. The propagation characteristics of optical fibers

with elliptic cross – section have been investigated by many researchers. Singh [13] have proposed an analytical

study of dispersion characteristics of helically cladded step – index optical fiber with elliptical core. Present

work is the study of circular optical waveguide with sheath helix in between the core and cladding region, this

work also gives the comparison of dispersion characteristic at different pitch angles. The sheath helix [12] is a


around the core – cladding boundary surfaces. As the number of propagating modes depends on the helix pitch

angle [2], so helical winding at core-cladding interface can control the dispersion characteristics [3-7] of the

optical waveguide. The winding angle of helix (ψ) can take any arbitrary value between 0 to π/2. In case of

sheath helix winding [1], cylindrical surface with high conductivity in the direction of winding which winds

helically at constant pitch angle (ψ) around the core cladding boundary surface. We assume that the waveguide

have real constant refractive index of core and cladding is n1 and n2 respectively (n1 > n2). In this type of optical

wave guide which we get after winding, the pitch angle controls the model characteristics of optical waveguide.

2. THEORETICAL BACKGROUND

The optical waveguide is the fundamental element that interconnects the various devices of an optical integrated

circuit, just as a metallic strip does in an electrical integrated circuit. However, unlike electrical current that

flows through a metal strip according to Ohm’s law, optical waves travel in the waveguide in distinct optical

modes. 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. The mode theory [10] is used to describe the properties of light that ray theory is unable to

explain. The mode theory uses electromagnetic wave behavior to describe the propagation of light along a fiber.

A set of guided electromagnetic waves is called the modes [13, 16] of the fiber. For a given mode, a change in

wavelength can prevent the mode from propagating along the fiber. The mode is no longer bound to the fiber.

The mode is said to be cut off [13]. Modes that are bound at one wavelength may not exist at longer

wavelengths. The wavelength at which a mode ceases to be bound is called the cutoff wavelength [11] for that

mode. However, an optical fiber is always able to propagate at least one mode. This mode is referred to as the

fundamental mode [16] of the fiber. The fundamental mode can never be cut off. We can take a case of a fiber

with circular cross-section wound with a sheath helix at the core-clad interface (Figure 1). A sheath helix can be

xxx

assumed by winding a very thin conducting wire around the cylindrical surface so that the spacing between the

nearest two windings is very small and yet they are insulated from each another. In our structure, the helical

windings are made at a constant helix pitch angle (ψ). We assume that (n1-n2) / n1 << 1.

3. WAVEGUIDE WITH CONDUCTING HELICAL WINDING

We consider the case of a fiber with circular cross – section wrapped with a sheath helix at core – clad boundary

as shown in Figure 1.

Figure 1: Fiber with circular cross – section wrapped with a sheath helix

In our structure, the helical windings are made at a constant angle ψ – the helix pitch angle. The structure has

high conductivity in a preferential direction. The pitch angle can control the propagation behavior of such fibers

[23]. We assume that the core and cladding regions have the real refractive indices n1 and n2 (n1 > n2), and (n1-

n2) / n1 << 1. The winding is right – handed and the direction of propagation is positive z direction. The winding

angle of the helix (pitch angle - ψ) can take any arbitrary value between 0 to π/2. This type of fibers is referred

to as circular helically cladded fiber (CHCF). This analysis requires the use of cylindrical coordinate system

( , , )r z [18] with the z – axis being the direction of propagation.

4. BOUNDARY CONDITIONS

Tangential component of the electric field in the direction of the conducting winding should be zero, and in the

direction perpendicular to the helical winding, the tangential component of both the electric field and magnetic

field must be continuous, so we have following boundary condition [17] with helix.



1 2 1 2 0z zE E cos E E sin (3)

1 2 1 2 0z zH H sin H H cos (4)

5. MODEL EQUATION

The guided mode along this type of fiber can be analyzed in a standard way, with the cylindrical coordinates

system ( , , )r z . In order to have a guided field the following conditions must be

satisfied2 2 1 1n k k k n k , where n1 and n2 are refractive indices or core and cladding regions

respectively. The solution of the axial field components can be written as,


xxxi

1 ( ) j j z j t

zE AJ ua e

(5)

1

j j z j t

zH BJ ua e

(6)


2 ( ) j j z j t

ZE CK ua e

(7)

2

j j z j t

zH DK ua e

(8)

where, , , ,A B C D are arbitrary constants which are to be evaluated from the boundary conditions. Also

J ua and ( )K wa are the Bessel functions.

For a guided mode, the propagation constant lies between two limits 2 and 1 . If 2 2 1 1n k k k n k

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 2

k

is free – space

propagation constant. The transverse field components can be obtained by using Maxwell’s standard relations.



1 2( ) '( ) j j z j tj


(9)

1 12( ) '( ) j j z j tj


(10)

The expressions for Eϕ and Hϕ inside the core are, when (r > a)

2 2( ) '( ) j j z j tj


(11)

2 22( ) '( ) j j z j tj


(12)

Now put these transverse field components equations into boundary conditions, we get following four unknown

equations involving four unknown arbitrary constants

2( ) sin cos '( ) cos 0

jAJ ua BJ ua

u a u

(13)

2( ) sin cos '( ) cos 0

jCK wa DK wa

w a w

(14)

2

2



jAJ ua BJ ua

u a u

jCK wa DK wa

w a w

(15)

xxxii

1

2

2

2



jAJ ua BJ ua

u u a

jCK wa DK wa

w w a

(16)

Equations (13), (14), (15) and (16) will yield a non – trivial solution if the determinant whose elements are the

coefficient of these unknown constants is set equal to zero. Thus we have

1 2 3 4

1 2 3 40

1 2 3 4

1 2 3 4

A A A A

B B B B

C C C C

D D D D

(17)

where,

21 ( ) sin cos

2 '( ) cos

3 0

4 0

A J uau a

jA J ua

u

A

A

(18)

2

1 0

2 0

3 ( ) sin cos

4 '( ) cos

B

B

B K waw a

jB K wa

w

(19)

2

2

1 ( ) cos sin

2 '( ) sin

3 ( ) cos sin

4 '( ) sin

C J uau a

jC J ua

u

C K waw a

jC K wa

w

(20)

xxxiii

1

2

2

2

1 '( ) cos

2 ( ) sin cos

3 '( ) cos

4 ( ) sin cos

jD J ua

u

D J uau a

jD K wa

w

D K waw a

(21)

After eliminating unknown constants from equations (17), (18), (19), (20) & (21), we get the following


1

2

2 2

2

2

2 2

2

2

( ) '( )sin cos cos

'( ) ( )


'( ) ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(22)

Equation (22) is standard characteristic equation, and is used for model dispersion properties and model cutoff

conditions.

6. SIMULATION RESULTS AND DISCUSSION

It is now possible to interpret the characteristic equation (Equation 22) in numerical terms. This will give us an

insight into model properties of our waveguide.

1

2

2 2

2

2

2 2

2

2

( ) '( )sin cos cos

'( ) ( )


'( ) ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(23)

2 2 2

2

2 2

1 2

( / )k nawb

V n n

(24)

2

2 2 2 2 2 2

1 2

2( ) ( )

aV u w a n n

(25)

where, b & V are known as normalization propagation constant & normalized frequency parameter respectively.

We make some simple calculations based on Equations (24) and (25). We choose n1=1.50, n2=1.46 and λ

=1.55µm. We take 1 for simplicity, but the result is valid for any value of .

In order to plot the dispersion relations, we plot the normalized frequency parameter V against the normalization

propagation constant b. we considered five special cases corresponding to the values of pitch angle ψ as 00, 30

0,

450, 60

0 and 90

0.

xxxiv

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b

Figure 2: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 00

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


xxxv

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b

Figure 5: Dispersion Curve of normalized propagation constant b as a function of V

for a lower – order modes for pitch angle ψ = 600

xxxvi

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


From the above figures we observe that, they all have standard expected shape, but except for lower order

modes they comes in pairs, that is cutoff values for two adjacent mode converge. This means that one effect of

conducting helical winding is to split the modes and remove a degeneracy which is hidden in conventional


We also observe that another effect of the conducting helical winding is to reduce the cutoff values, thus

increasing the number of modes. This effect is undesirable for the possible use of these waveguide for long

distance communication.


0 and 60


near the lowest order mode. It is found that on the left of the lowest cutoff values, portions of curves appear

which have no resemblance with standard dispersion curves, and have no cutoff values. This means that for very

small value of V anomalous dispersion properties may occur in helically wound waveguides.

We found that some curves have band gaps of discontinuities between some value of V. These represent the

band gaps or forbidden bands of the structure. These are induced by the periodicity of the helical windings.

7. CONCLUSION

From the above results we observe that, the effect of the conducting helical winding is to reduce the cutoff

values, thus increasing the number of modes. This effect is undesirable for the possible use of these waveguide

for long distance communication.

We also observe that, all curves have standard expected shape, but except for lower order modes they comes in

pairs, that is cutoff values for two adjacent mode converge. This means that one effect of conducting helical

winding is to split the modes and remove a degeneracy which is hidden in conventional waveguide without

windings.


0 and 60





xxxvii


band gaps or forbidden bands of the structure. These are induced by the periodicity of the helical windings. Thus

helical pitch angle controls the modal properties of this type of optical waveguide.

REFRENCES

[1] Kumar, D. and O. N. Singh II, “Modal characteristics equation and dispersion curves for an elliptical

step-index fiber with a conducting helical winding on the core-cladding boundary - An analytical

study,” IEEE, Journal of Light Wave Technology, Vol. 20, No. 8, 1416–1424, USA, August 2002.


[3] V.N. Mishra, Vivek Singh, B. Prasad, S. P. Ojha, “Optical Dispersion curves of two metal - clad

lightguides having double convex lens core cross sections”, Wiley, Microwave and Optical Technology

Letters, Vol. 24, No. 4, 229-232, New York, Feb 20, 2000.

[4] V.N. Mishra, V. Singh, B. Prasad, S. P. Ojha, “An Analytical investigation of dispersion characteristic of

a lightguide with an annular core cross section bounded by two cardioids”, Wiley, Microwave and


[5] V. Singh, S. P. Ojha, B. Prasad, and L. K. Singh, “Optical and microwave Dispersion curves of an optical

waveguide with a guiding region having a core cross section with a lunar shape”, Optik 110, 267-270,

1999.

[6] V. Singh, S. P. Ojha, and L. K. Singh, “Model Behaviour, cutoff condition, and dispersion characteristics

of an optical waveguide with a core cross section bounded by two spirals”, microwave Optical


[7] V. Singh, S. P. Ojha, and B. Prasad, “weak guidance modal dispersion characteristics of an optical

waveguide having core with sinusoidally varying gear shaped cross section”, microwave Optical


[8] Gloge D., “Dispersion in weakly guiding fibers,” Appl. Optics, Vol. 10, 2442 - 2445, 1971.

[9] P. K. Choudhury, D. Kumar, and Z. Yusoff, F. A. Rahman, “An analytical investigation of four-layer

dielectric optical fibers with au nano-coating - A comparison with three-layer optical fibers”, PIER 90,

269 - 286, 2009.


edition McGraw-Hill,Singapore,2000.


[12] Kumar, D. and O. N. Singh II, “Towards the dispersion relations for dielectric optical fibers with helical

windings under slow and fast wave considerations – a comparative analysis,” PIER, Vol. 80, 409–420,

2008.

[13] Kumar, D. and O. N. Singh II, “An analytical study of the modal characteristics of annular step – index

fiber of elliptical cross – section with two conducting helical windings on the two boundary surfaces

between the guiding and non – guiding regions” Optik, Vol. 113, No. 5, 193-196, 2002.

[14] Singh, U. N., O. N. Singh II, P. Khastgir and K. K. Dey “Dispersion characteristics of helically cladded

step – index optical fiber analytical study” J. Opt. Soc. Am. B, 1273-1278, 1995.

[15] M. P. S. Rao, Vivek Singh, B. Presad and S. P. Ojha “Model characteristic and dispersion curves of

hypocycloidal optical waveguide” Optik, 110, No. 2, 81-85, 1999.

[16] Ajoy Ghatak and K. Thyagarajan, “Optical Electronics” Cambridge University Press, India, 2008.

[17] Kumar, D. and O. N. Singh II, “Some special cases of propagation characteristics of an elliptical step –

index fiber with a conducting helical winding on the core – cladding boundary – An analytical treatment

,” Optik Vol. 112, No. 12, 561-566, 2001.

xxxviii

[18] Govind P. Agrawal, “Fiber – Optic Communication Systems”, 3rd

edition, A John Wiley & Sons, Inc.,

Publication, New York, 2002.

xxxviii

DISPERSION & CUTTOFF CHARACTERISTICS OF CIRCULAR

HELICALLY CLADDED OPTICAL FIBER

V. Mishra1, 2

, A. K. Gautam1, B. R. Taunk

1

1Electronics Engineering Department, Sardar Vallabhbhai National Institute of Technology Surat, India

2Sr. Member IEEE

Email: [email protected]

ABSTRACT

In this article dispersion characteristic of conventional optical waveguide with helical winding at core –

cladding interface has been obtained. The model dispersion characteristics of optical waveguide with helical

winding at core-cladding interface have been obtained for five different pitch angles. This article includes

dispersion characteristics of optical waveguide with helical winding, and compression of dispersion

characteristics of optical waveguide with helical winding at core-cladding interface for five different pitch

angles. Boundary conditions have been used to obtain the dispersion characteristics and these conditions have

been utilized to get the model Eigen values equation. From these Eigen value equations dispersion curve are

obtained and plotted for modified optical waveguide for particular values of the pitch angle of the winding and

the effect of this winding has been discussed. We observe that the effect of conducting helical winding is to

reduce the cutoff values, thus increasing the number of modes; we also observe that for very small value of V

anomalous dispersion properties may occur in helically wound waveguides. We found that some curves have

band gaps of discontinuities between some value of V. Thus helical pitch angle controls the modal properties of

this type of optical waveguide.

Keywords: dispersion curves, Bessel functions, characteristics equation, sheath helix, circular waveguide,

modal cutoff, CHCF.

INTRODUCTION

An optical waveguide is basically a cylindrical dielectric waveguide with a circular cross section where a high-

index wave guiding core is surrounded by a low-index cladding. The index step and profile are controlled by the

concentration and distribution of dopants. Silica fibers are ideal for light transmission in the visible and near-

infrared regions because of their low loss and low dispersion in these spectral regions. They are therefore

suitable for optical communications. Even though optical fiber seems quite flexible, it is made of glass, which

cannot withstand sharp bending or longitudinal stress. Therefore when fiber is placed inside complete cables

special construction techniques are employed to allow the fiber to move freely within a tube. Usually fiber optic

cables contain several fibers, a strong central strength member and one or more metal sheaths for mechanical

protection. Some cables also include copper pairs for auxiliary applications. Optical fibers with helical winding

are known as complex optical waveguides. The use of helical winding in optical fibers makes the analysis much

accurate. As the number of propagating modes depends on the helix pitch angle, so helical winding at core –

cladding interface can control the dispersion characteristics of the optical waveguide [3].

The conventional optical fiber having a circular core cross – section which is widely used in optical

communication systems [1]. Recently metal – clad optical waveguides have been studied because these provide

potential applications, connecting the optical components to other circuits. Metallic – cladding structure on an

optical waveguide is known as a TE – mode pass polarizer and is commercially applied to various optical

devices [4]. The propagation characteristics of optical fibers with elliptic cross – section have been investigated

by many researchers. Singh [5] have proposed an analytical study of dispersion characteristics of helically

cladded step – index optical fiber with circular core. The model characteristic and dispersion curves of a

hypocycloidal optical waveguide have been investigated by Ojha [6]. Present work is the study of circular

optical waveguide with sheath helix [3] in between the core and cladding region. The sheath helix is a


around the core – cladding boundary surfaces.

Optical fibers with helical winding are known as complex optical waveguides. The conventional optical fiber

having a circular core cross – section which is widely used in optical communication systems. The use of helical

winding in optical fibers makes the analysis much accurate [1]. The propagation characteristics of optical fibers

with elliptic cross – section have been investigated by many researchers. Singh [13] have proposed an analytical

xxxix

study of dispersion characteristics of helically cladded step – index optical fiber with elliptical core. Present

work is the study of circular optical waveguide with sheath helix in between the core and cladding region, this

work also gives the comparison of dispersion characteristic at different pitch angles. The sheath helix [12] is a


around the core – cladding boundary surfaces. As the number of propagating modes depends on the helix pitch

angle [2], so helical winding at core-cladding interface can control the dispersion characteristics [3-7] of the

optical waveguide. The winding angle of helix (ψ) can take any arbitrary value between 0 to π/2. In case of

sheath helix winding [1], cylindrical surface with high conductivity in the direction of winding which winds

helically at constant pitch angle (ψ) around the core cladding boundary surface. We assume that the waveguide

have real constant refractive index of core and cladding is n1 and n2 respectively (n1 > n2). In this type of optical

wave guide which we get after winding, the pitch angle controls the model characteristics of optical waveguide.

THEORETICAL BACKGROUND

The optical waveguide is the fundamental element that interconnects the various devices of an optical integrated

circuit, just as a metallic strip does in an electrical integrated circuit. However, unlike electrical current that

flows through a metal strip according to Ohm’s law, optical waves travel in the waveguide in distinct optical

modes. 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. The mode theory [10] is used to describe the properties of light that ray theory is unable to

explain. The mode theory uses electromagnetic wave behavior to describe the propagation of light along a fiber.

A set of guided electromagnetic waves is called the modes [13, 16] of the fiber. For a given mode, a change in

wavelength can prevent the mode from propagating along the fiber. The mode is no longer bound to the fiber.

The mode is said to be cut off [13]. Modes that are bound at one wavelength may not exist at longer

wavelengths. The wavelength at which a mode ceases to be bound is called the cutoff wavelength [11] for that

mode. However, an optical fiber is always able to propagate at least one mode. This mode is referred to as the

fundamental mode [16] of the fiber. The fundamental mode can never be cut off. We can take a case of a fiber

with circular cross-section wound with a sheath helix at the core-clad interface (Figure 1). A sheath helix can be

assumed by winding a very thin conducting wire around the cylindrical surface so that the spacing between the

nearest two windings is very small and yet they are insulated from each another. In our structure, the helical

windings are made at a constant helix pitch angle (ψ). We assume that (n1-n2) / n1 << 1.

WAVEGUIDE WITH CONDUCTING HELICAL WINDING

We consider the case of a fiber with circular cross – section wrapped with a sheath helix at core – clad boundary

as shown in Figure 1.

Figure 1: Fiber with circular cross – section wrapped with a sheath helix

In our structure, the helical windings are made at a constant angle ψ – the helix pitch angle. The structure has

high conductivity in a preferential direction. The pitch angle can control the propagation behavior of such fibers

[23]. We assume that the core and cladding regions have the real refractive indices n1 and n2 (n1 > n2), and (n1-

n2) / n1 << 1. The winding is right – handed and the direction of propagation is positive z direction. The winding

angle of the helix (pitch angle - ψ) can take any arbitrary value between 0 to π/2. This type of fibers is referred

xl

to as circular helically cladded fiber (CHCF). This analysis requires the use of cylindrical coordinate system

( , , )r z [18] with the z – axis being the direction of propagation.

BOUNDARY CONDITIONS

Tangential component of the electric field in the direction of the conducting winding should be zero, and in the

direction perpendicular to the helical winding, the tangential component of both the electric field and magnetic

field must be continuous, so we have following boundary condition [17] with helix.



1 2 1 2 0z zE E cos E E sin (3)

1 2 1 2 0z zH H sin H H cos (4)

MODEL EQUATION

The guided mode along this type of fiber can be analyzed in a standard way, with the cylindrical coordinates

system ( , , )r z . In order to have a guided field the following conditions must be

satisfied2 2 1 1n k k k n k , where n1 and n2 are refractive indices or core and cladding regions

respectively. The solution of the axial field components can be written as,


1 ( ) j j z j t

zE AJ ua e

(5)

1

j j z j t

zH BJ ua e

(6)


2 ( ) j j z j t

ZE CK ua e

(7)

2

j j z j t

zH DK ua e

(8)

where, , , ,A B C D are arbitrary constants which are to be evaluated from the boundary conditions. Also

J ua and ( )K wa are the Bessel functions.

For a guided mode, the propagation constant lies between two limits 2 and 1 . If 2 2 1 1n k k k n k

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 2

k

is free – space

propagation constant. The transverse field components can be obtained by using Maxwell’s standard relations.



1 2( ) '( ) j j z j tj


(9)

xli

1 12( ) '( ) j j z j tj


(10)

The expressions for Eϕ and Hϕ inside the core are, when (r > a)

2 2( ) '( ) j j z j tj


(11)

2 22( ) '( ) j j z j tj


(12)

Now put these transverse field components equations into boundary conditions, we get following four unknown

equations involving four unknown arbitrary constants

2( ) sin cos '( ) cos 0

jAJ ua BJ ua

u a u

(13)

2( ) sin cos '( ) cos 0

jCK wa DK wa

w a w

(14)

2

2



jAJ ua BJ ua

u a u

jCK wa DK wa

w a w

(15)

1

2

2

2



jAJ ua BJ ua

u u a

jCK wa DK wa

w w a

(16)

Equations (13), (14), (15) and (16) will yield a non – trivial solution if the determinant whose elements are the

coefficient of these unknown constants is set equal to zero. Thus we have

1 2 3 4

1 2 3 40

1 2 3 4

1 2 3 4

A A A A

B B B B

C C C C

D D D D

(17)

where,

21 ( ) sin cos

2 '( ) cos

3 0

4 0

A J uau a

jA J ua

u

A

A

(18)

xlii

2

1 0

2 0

3 ( ) sin cos

4 '( ) cos

B

B

B K waw a

jB K wa

w

(19)

2

2

1 ( ) cos sin

2 '( ) sin

3 ( ) cos sin

4 '( ) sin

C J uau a

jC J ua

u

C K waw a

jC K wa

w

(20)

1

2

2

2

1 '( ) cos

2 ( ) sin cos

3 '( ) cos

4 ( ) sin cos

jD J ua

u

D J uau a

jD K wa

w

D K waw a

(21)

After eliminating unknown constants from equations (17), (18), (19), (20) & (21), we get the following


1

2

2 2

2

2

2 2

2

2

( ) '( )sin cos cos

'( ) ( )


'( ) ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(22)

Equation (22) is standard characteristic equation, and is used for model dispersion properties and model cutoff

conditions.

SIMULATION RESULTS AND DISCUSSION

It is now possible to interpret the characteristic equation (Equation 22) in numerical terms. We now make some

simple calculations based on equation (23), equation (24) and equation (25). This will give us an insight into

model properties of our waveguide.

1

2

2 2

2

2

2 2

2

2

( ) '( )sin cos cos

'( ) ( )


'( ) ( )

kJ ua J uau

J ua u a u J ua

kK wa K waw

K wa w a w K wa

(23)

xliii

2 2 2

2

2 2

1 2

( / )k nawb

V n n

(24)

2

2 2 2 2 2 2

1 2

2( ) ( )

aV u w a n n

(25)

where, b & V are known as normalization propagation constant & normalized frequency parameter respectively.

We make some simple calculations based on Equations (24) and (25). We choose n1=1.50, n2=1.46 and λ

=1.55µm. We take 1 for simplicity, but the result is valid for any value of .

In order to plot the dispersion relations, we plot the normalized frequency parameter V against the normalization

propagation constant b. we considered five special cases corresponding to the values of pitch angle ψ as 00, 30

0,

450, 60

0 and 90

0.

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


xliv

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b



xlv

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b


0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

b



From the above figures we observe that, they all have standard expected shape, but except for lower order

modes they comes in pairs, that is cutoff values for two adjacent mode converge. This means that one effect of






xlvi


0 and 60






band gaps or forbidden bands of the structure. These are induced by the periodicity of the helical windings.

We now come to table 1. we note particularly that the dependence of the cutoff V – value (Vc) on ψ is such that

as ψ is increased there is a drastic fall in Vc at ψ =300 and then a small increase as ψ goes from 30

0 to 60

0; then is

a quick rise as ψ changes from 600 to 90

0 (Figure 7).

Table 1: Cutoff Vc values for some lower – order modes


00 1.80 3.80 4.00 6.90 7.10 10.10 10.30 - -

300 0.05 1.70 1.80 3.70 3.90 7.00 7.10 10.20 10.30

450 0.40 1.70 1.80 3.65 3.70 7.00 7.20 10.20 10.30

600 0.30 1.50 1.80 3.70 3.90 7.00 7.20 10.20 10.30

900 1.90 3.80 5.40 7.00 8.60 10.20 11.80 - -

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

Angle in Degree

Vc

Figure 7: Dependence of cutoff values Vc on the pitch angle ψ

xlvii

Thus the two most sensitive regions in respect of the influence of helical pitch angle ψ on the cutoff values and

the model properties of waveguides are ranges from ψ = 00 to ψ = 30

0 and ψ = 60

0 to ψ = 90

0 and these ranges of

pitch angle expected to have potential applications with ψ as a means for controlling the model properties.

CONCLUSION

From the above results we observe that, they all have standard expected shape, but except for lower order modes

they comes in pairs, that is cutoff values for two adjacent mode converge. This means that one effect of







0 and 60






band gaps or forbidden bands of the structure. These are induced by the periodicity of the helical windings. Thus

helical pitch angle controls the modal properties of this type of optical waveguide.

REFRENCES

[1] Kumar, D. and O. N. Singh II, “Modal characteristics equation and dispersion curves for an elliptical

step-index fiber with a conducting helical winding on the core-cladding boundary - An analytical

study,” IEEE, Journal of Light Wave Technology, Vol. 20, No. 8, 1416–1424, USA, August 2002.


[3] V.N. Mishra, Vivek Singh, B. Prasad, S. P. Ojha, “Optical Dispersion curves of two metal - clad

lightguides having double convex lens core cross sections”, Wiley, Microwave and Optical Technology

Letters, Vol. 24, No. 4, 229-232, New York, Feb 20, 2000.

[4] V.N. Mishra, V. Singh, B. Prasad, S. P. Ojha, “An Analytical investigation of dispersion characteristic of

a lightguide with an annular core cross section bounded by two cardioids”, Wiley, Microwave and


[5] V. Singh, S. P. Ojha, B. Prasad, and L. K. Singh, “Optical and microwave Dispersion curves of an optical

waveguide with a guiding region having a core cross section with a lunar shape”, Optik 110, 267-270,

1999.

[6] V. Singh, S. P. Ojha, and L. K. Singh, “Model Behaviour, cutoff condition, and dispersion characteristics

of an optical waveguide with a core cross section bounded by two spirals”, microwave Optical


[7] V. Singh, S. P. Ojha, and B. Prasad, “weak guidance modal dispersion characteristics of an optical

waveguide having core with sinusoidally varying gear shaped cross section”, microwave Optical


[8] Gloge D., “Dispersion in weakly guiding fibers,” Appl. Optics, Vol. 10, 2442 - 2445, 1971.

[9] P. K. Choudhury, D. Kumar, and Z. Yusoff, F. A. Rahman, “An analytical investigation of four-layer

dielectric optical fibers with au nano-coating - A comparison with three-layer optical fibers”, PIER 90,

269 - 286, 2009.


edition McGraw-Hill,Singapore,2000.

xlviii


[12] Kumar, D. and O. N. Singh II, “Towards the dispersion relations for dielectric optical fibers with helical

windings under slow and fast wave considerations – a comparative analysis,” PIER, Vol. 80, 409–420,

2008.

[13] Kumar, D. and O. N. Singh II, “An analytical study of the modal characteristics of annular step – index

fiber of elliptical cross – section with two conducting helical windings on the two boundary surfaces

between the guiding and non – guiding regions” Optik, Vol. 113, No. 5, 193-196, 2002.

[14] Singh, U. N., O. N. Singh II, P. Khastgir and K. K. Dey “Dispersion characteristics of helically cladded

step – index optical fiber analytical study” J. Opt. Soc. Am. B, 1273-1278, 1995.

[15] M. P. S. Rao, Vivek Singh, B. Presad and S. P. Ojha “Model characteristic and dispersion curves of

hypocycloidal optical waveguide” Optik, 110, No. 2, 81-85, 1999.

[16] Ajoy Ghatak and K. Thyagarajan, “Optical Electronics” Cambridge University Press, India, 2008.

[17] Kumar, D. and O. N. Singh II, “Some special cases of propagation characteristics of an elliptical step –

index fiber with a conducting helical winding on the core – cladding boundary – An analytical treatment

,” Optik Vol. 112, No. 12, 561-566, 2001.

[18] Govind P. Agrawal, “Fiber – Optic Communication Systems”, 3rd

edition, A John Wiley & Sons, Inc.,

Publication, New York, 2002.

Documents

DISPERSION & CUTOFF CHARACTERISTICS OF CIRCULAR …ajaykmga.weebly.com/uploads/2/2/1/5/2215854/m._tech_defence.pdf · important moments of making right decisions and I am glad to