Analytical Analysis of Ray Characteristics Inside the Optical Fiber

JOURNAL OF TELECOMMUNICATIONS, VOLUME 18, ISSUE 1, JANUARY 2013

© 2012 JOT www.journaloftelecommunications.co.uk

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Analytical Analysis of ray characteristics inside the optical fiber

*Chakresh Kumar,**Girish Narah and **Aroop Sharma

Abstract-In this paper we obtain solution of the ray equations in a parabolic and elliptical refractive index profile. We studied the various

conditions for the suitable propagation of ray inside the fiber. A comparison is also made between the ray propagating in elliptical and

parabolic refractive index profile.

Keywords :- Graded index fibre, Ray theory concept, refractive index profile

INTRODUCTION:- We consider a medium of varying refractive

index n=n(x). According to snell’s law

n1sinΦ1=n2sinΦ2=constant ,Φ1, Φ2, are the angle of

incidence at various interfaces If Θ1, Θ2, ........are the

corresponding angles that the ray makes with z- axis

Then n1cosθ1=n2cosθ2=n3cosθ3=constant (β) (a)

When the refractive index variation is continuous, the

thickness of each layer becomes infinitesimally small

and it form a continuous curve as shown in above

figure and it is taken form reference [1]

n(x)cosθ(x)=β(invariant of ray path)

1)()(

(dz) + (dx) = (ds)

22

222

dzdxdzds

or

Cosθ = (ds

dz)

-1 =dz

ds

We obtain

(ds

dz)=

1

cos θ x =

n x

β[from eqn (a)]

Substituting ds

dzin eqn (b)

(dx

dz)

2= 1

)(2

2

xn

(c)

Differentiating eqn(c) with respect to z

dz

dx

dx

(x)dn1=

dz

xd

dz

dx2

2

22

2

dx

xdn

dz

xd )(

2

1 2

22

2

(1)

The above equation is another form of ray equation

Ray Path In Parabolic Refractive Index

The parabolic refractive index is characterised by the

following refractive index distribution

])(21[)( 22

1

2

a

xnxn When | x|< a (core)

=2

2

2

1 ]21[ nn when |x|>a (cladding) (2)

Applying equation (2) in equation (1), we get

])(21[2

1 22

122

2

a

xn

dx

d

dz

xd

= )}({2

22

2

1 zxa

n

*Assistant Professor Electronics and communication department Tezpur (central) University,Tezpur,Assam India

** UG studentsElectronics and communication department Tezpur (central) University,Tezpur,Assam India



14

)(2

2

2

zxdz

xd , Where

1

2n

a

)(2

2

2

zxdz

xd =0

Therefore the general solution is given by

x(z)=AsinГz + BcosГz (3)

and similarly we can find

y(z)=CsinГz + Dcos Гz (4)

Where A, B,C & D are the Constants which can be

determined by the initial launching condition of the

ray.Now, let us consider that an extreme form of

skew rays is launched on the x-axis(at x= a’) in the y-z

plane(making angle θ’ with the z-axis). Thus , the

launching conditions on plane z=0 are

xz=0=a’ =>B=a’

And 𝑑𝑥

𝑑𝑧|z=0=0 =>A=0

Putting the value of A and B in equation (3), we get :-

Thus x(z)=a’cosГz (5)

And

Yz=0=0 =>D=0

And 𝑑𝑦

𝑑𝑧|z=0= tanθ’

=>C=𝑡𝑎𝑛𝜃 ’

Г

=>C=2

'tana

1n

If at the launching point n=n’ then β=n’cosθ’, so

C=2

'sin'

1n

an

Putting the value of C and D in equation (4)

zn

anzy

sin

2

'sin')(

1

Now if an

na

2''sin 1 , then

zazy sin')( (6)

Suppose a=50μm, a’=20μm,Δ=0.04, θ’=15˚ the

propagation of the ray is shown in fig (1)

Fig (1)

Keeping the other factor fixed and varying θ’ such as

θ’=45˚ and θ’=90˚, we observe the propagation in fig

(2) and fig(3) respectively

Fig (2)

Fig (3)

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Helical ray propagation

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The number of helical turn’s increases as the angle is

varied from 0 ˚ to 90˚ that means the ray paths get

denser as the angle is increased.

Suppose a=10μm, a’=20μm, Δ=0.04, θ=60˚ the

propagation of the ray is shown in fig (4)

Fig (4)

Now keeping the other terms constant and varying

core radius a such that a=20μm and a=60μm, the

propagation is shown in fig(5) and fig(6)

Fig (5)

Fig (6)

The number of helical turn’s decreases as the core

radius is increased as observed from the graphs in fig

(4),(5),(6)

Now in fig (4), if we change the value of Δ such that

Δ=0.03 and Δ=0.07, the propagation will be shown in

fig (7) and fig (8)

Fig (7)

Fig (8)

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The number of helical turn’s increases as the value of

Δ is increased as observed from fig (4),(7),(8)Suppose

a=50μm, a’=5μm, Δ=0.04, θ=60˚ the propagation will

be as shown in fig (9)

Fig (9)

Keeping the other terms constant and changing a’

such that a’=10μm and a’=60μm, the propagation is

shown in fig (10) and fig(11)

Fig (10)

Fig (11)

The radius of helical turns increases as the launching

point is increased, as observed from the figure

(9),(10),(11)

Ray Path In Elliptical Refractive Index

Now we will obtain the ray paths in an elliptical

index fibre characterized by the following refractive

9.5

Fig (12)

Here a=2b. For 4a=b the propagation is shown in fig

(13)

Fig (13)

As clear from fig(12) and fig(13) there is no fixed

shape for the ray path in elliptical refractive index.

When a =b the elliptical refractive index behaves as

parabolic refractive index as show in fig(14)

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ray propagation in elliptical index

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Fig (14)

CONCLUSION

In this paper ray characteristics in parabolic and

elliptical index profile fiber has been studied at

different parameter.We observed that the ray graphs

in the parabolic refractive index by varying core

radius (a), angle (θ'), relative core-cladding index

difference (Δ) and launching point(a’) . We conclude

that-

Keeping θ', Δ, a' constant and

varying core radius (a) we found

that the no. of helical path decreases

with increase in core radius.

Keeping a, Δ, a' constant and

varying θ' from 0˚ to 180˚ we found

that the no. of helical path increases

when angle is increased from 0˚ to

90˚ after that the no. of helical path

decreases with the increase in angle

after 90˚ to 180˚. No of helical path is

maximum at 90˚.

Keeping a, a', θ' constant and

varying Δ we found that the no. of

helical path increases with increase

in Δ.

Keeping a, θ', Δ constant and

varying launching point (a') we

found that the radius of each helical

path increases with increase in the

launching point.

Increase in the no. of helical path and radius of the

helical path result in internal time delay i.e. time

required to send data through the optical fiber will

increase with increase in angle from 0˚ to 90˚,

increase in Δ and increase in launching point (a') and

since no. of helical path decreases with increase in

core radius, therefore to avoid time delay the value of

angle (θ'), Δ and a' should be as less as possible and

core radius should be high.

References

1. Ajoy Ghatak, K. Thyagarajan,(1999).

Introduction to Fiber Optics,Cambridge University

press.

2. Checcacci, P. F.,(1980). “Applicability of an

asymptotic numerical method to the

determination of whispering and bouncing

modes in elliptical fibers,” Journal of the Optical

Society of America vol. 70 no. 12.

3. Pierre Aschiéri,(2006). “Complex behavior of a

ray in a Gaussian index profile periodically

segmented waveguide ” J. Opt. A: Pure Appl.

Opt. vol. 8 pp. 386.

4. Hagen Renner, (1998). “Polarization

characteristics of optical waveguides with

separable symmetric refractive-index profiles,”

Journal of the Optical Society of America A vol.

15, no. 5.

5.S.K RaghuwanshiRay Paths In An Elliptic

Parabolic Refractive Index Profile FiberWorld

Journal of Science and Technology 2011, 1(8): 74-

78ISSN: 2231 – 2587

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Documents

Analytical Analysis of Ray Characteristics Inside the Optical Fiber