DEPENDENCE OF CUTOFF VALUES ON

PITCH ANGLE OF CIRCULAR HELICALLY

CLADDED OPTICAL WAVEGUIDE AT ITS

CORE CLADDING INTERFACE Ajay Kumar Gautam

1, Amit Kumar Katariya

2, Dr. Vivekanand Mishra

3

1ajaysvnit@gmail.com, 2er.amitkatariya@gmail.com, 3vive@eced.svnit.ac.in 1, 2

Dev Bhoomi Institute of Technology, Dehradun, Uttarakhand 3Sr. Member IEEE

3Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat

Abstract- This article includes dispersion characteristics of

optical waveguide with helical windings, 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.

I. 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 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 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. 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.

III. 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 (1)

2 20

zE sin E cos (2)

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. MODAL EQUATIONS

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

2k

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 tjE j AJ ua uBJ ua e

u a

(9)

1 12( ) '( )

j j z j tjH j BJ ua uAJ ua e

u a

(10)

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

2 2( ) '( )

j j z j tjE j CK wa wDK wa e

w a

(11)

2 22( ) '( )

j j z j tjH j DK wa wCK wa e

w a

(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

( ) cos sin '( ) sin

( ) cos sin '( ) sin 0

jAJ ua BJ ua

u a u

jCK wa DK wa

w a w

(15)

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

(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)

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)

Evaluation of the above determinant yields the following Eigen

value 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

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

After eliminating unknown constants from equations (17),

(18), (19), (20) & (21), 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

(22)

Equation (22) is standard characteristic equation, and is used

for model dispersion properties and model cutoff conditions.

V. RESULTS

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

'( ) ( )

( ) '( )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

(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, 45

0, 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

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

Figure 3: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 300

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 4: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 450

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

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 6: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 900

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 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.

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 300 to 60

0; then is a quick rise as

ψ changes from 600 to 90

0 (Figure 7).

TABLE I

CUTOFF VC VALUES FOR SOME LOWER ORDER MODES

ψ 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

Figure 7: 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 = 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.

VI. 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 modes 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.

An anomalous feature in the dispersion curves is observable

for ψ = 300, 45

0 and 60

0 for this type of waveguide 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.

REFERENCES

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