Ch2A Mode Theory Fundamentals

8/2/2019 Ch2A Mode Theory Fundamentals

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Optical fibre waveguides:

Optical fibres are one of the most important components in an optical fibre system because

its transmission characteristics play an important role in the performance of the entire system.

Some important questions on optical fibre:

what is the structure of an optical fibre?

how does light propagate along the fibre?

what materials are fibres made of?

how is it fabricated?

what is the signal loss or attenuation mechanism in a fibre?

Optical Fibre Waveguide


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How does light propagate along a fibre?

Need to understand the physics of the problem.

Basically, the involved theory can be divided into two types as follows:

Ray theory transmissionWave theory transmission

Important definition:

Refractive index n is the the ratio of the speed of light in vacuum to that in the material.n = c/v

Upon entering a material, the speed of light would reduce, and it depends on the type of

material.

The concept of reflection and refraction can be understood by considering the behaviour of

light rays travelling in a dielectric material.

When light ray encounters a boundary separating two different media, a part of the ray is

reflected back ( ifn1 > n2) and the remainder will bend (refracted) as it enters the second

medium.



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Snells law states the relationship between the refractive indices of materials, and

the angles of incidence and refraction as:

n1sin1 = n2sin2

where n1 and n2 are the refractive indices of different materials, while 1 and 2 are

the angles of incidence and refraction, respectively.

RAY THEORY TRANSMISSION

Snells Law:


21

Normal

1

2


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Critical AngleAs n1 > n2, the angle of refraction is always greater than the angle of incidence.

When the angle of refraction 2 is 90o (parallel to the interface), the incidence angle is called

the critical angle c.

Snells law becomes: sin c = n2 / n1Total Internal Reflection:At the angles of incidence greater than the critical angle, light is reflected back into the originating

dielectric medium.

Total internal reflection occurs when:

n1 > n2.

angle of incidence exceeds the critical angle.

This is the basic mechanism by which light can propagate down an optical fibre.



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High Index n1

(glass)

Incident Ray

Partial Internal

Reflection

Exit RayLow Index n2

(air)

1

2

c

2

1 2



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Transmission of a light ray in a perfect optical fibre:



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Acceptance Angle

We have considered the propagation of light in an optical fibre. But what is the amount of light

which can enter the optical fibre?

Since only rays with a sufficiently shallow grazing angle at the interface is transmitted, it is

clear that not all the rays entering the fibre core will continue to propagate down the fibre.



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Numerical Aperture

The relationship between the acceptance angle and the refractive indices of the media leads to the

definition of numerical aperture NA, which is given as:

NA = nosin

a= (n

1

2 n2

2)1/2

It should be noted that rays over the range of 0 < 1 < a will be propagated within the fibre.

The NA can also be written in terms of the relative refractive index as:

NA = n1(2)1/2

where = (n12 n22)/ 2n12

or (n1 n2)/n1 forn1 > n2.



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Waveguide Theory

In the previous section, we have discussed the propagation of light in terms of

ray theory.

In order to obtain a more detail model for the EM wave propagation,electromagnetic wave theory must be considered.

The basis is provided by Maxwells equations:

t

BE

=

r

rr

0= Drr

0= Brr

D = E and B = H

EtEH +=

rr

(1)

(2)

(3)

(4)



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Substituting forD and B and taking the curl of eqs. (1) and (2)

( )2

2

t

EE

=

r

r

( )2

2

t

HH

=

r

r

These lead to the nondispersive wave equations in a nonconducting medium:

1=pvwhere is the phase velocity in the medium.

( ) ( ) EEE2

=

Vector identity:


2

2

22

22 1

t

H

vt

HH

p

=

=

rr

r

2

2

22

22 1

t

E

vt

EE

p

=

=

vv

r


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IfEand Hvary sinusoidally with time at the frequency of light, thenEand H

can be written as

E = Ex exp (jt) ax and H =Hy exp(jt) ay, then from eqs. (1) and (2)

Differentiating eq. (5) with respect to zand substituting from eq. (6) we get

(5) (6)


EEjzH +=Hjz

E =

EjEz

E +=

2

2

2

HjH

z

H +=

22

2


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Let 2 = 2 +j, the solution of these equations are:

E = Exoexp(jt) exp(z)

H = Hyoexp(jt) exp(z)

Writing = + j, being the attenuation coefficient and the phaseconstant, the electric and magnetic fields are given by

E=Exo cos(tz) exp(z)

H= Hyo cos(tz) exp(z)

These equations describe a TEM wave travelling in the positive z-

direction undergoing attenuation exp(z).

Useful expression:

exp(jt) = cos(t) + j sin(t)



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Rewriting the propagation constant as :

= + j and

2 = 2 + j

Equating the real and imaginary parts:

2 2 = 2

and 2 =

As glass is an insulator, the conductivity is very low, and the relative

permeability is approximately unity. Therefore, = 0 and r= 1. In thissituation, we get = 0 and = (). So, the fields are given by

E = Exo cos(tz) and H = Hyo cos(tz)



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Cartesian and cylindrical polar coordinates of the wave equations hold for

each component of the field vector, every component satisfying the scalar

wave equation.

represents a component of the E orH field.

For a scalar wave equation: ,

the solution is given by

wherek

is the propagation vector or wave number. It carries the informationabout the direction of propagation, and the rate of change of phase with

distance. The magnitude ofkis

2

2

2

2 1

tvp

=

r

rktjr

r

= exp0

2=k



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Phase velocity:

At a given time, say at t = 0 and z = 0, the amplitude of the wave is zero.

After t1 sec, i.e. at t = t1 the amplitude of the wave is still zero but it has

traveled a distance z1. Thus,

sin(t1 z1) = 0 i.e t1z1 = 0, which gives t1 = z1

The velocity of propagation is given by z1/t1 = /, which is called phasevelocity vp.

Thus, the phase velocity is the velocity of propagation of a point of constantphase in the wave.

vp = / = /() = 1/()

If the dielectric medium is free space, then vp is the velocity of light c = 3108

m/s. For a dielectric of relative permitivity r, the phase velocity is given by

vp = 1/(0r)= c/ r= c/n

where n = r is the refractive index of the dielectric medium.



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If0 is the wavelength in the free space, the wavelength inthe dielectric material is given by

= 2/ = 2vp/ = 2c/n = c/nv = 0/n

Thus, we get an alternative definition of given by

= 2/ = 2n/0 = nk

where k = 2/0 is the propagation constant in the free

space.



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Group velocity:

In practice, the light do not travel with only one wavelength, but instead, a

combination of wavelengths.

The light is generally composed of a group of plane wave components

resulting in a packet of waves.

This wave packet do not travel at the phase velocity of individual waves, but

propagate with a group velocity vg.

1/vg = d/d = (d/d)( d/d)

From above = 2/ = 2vp/ = 2c/n , so that d/d = 2c/n2 = /

d/d = /, and d/d = d/d{2n1/) = 2n1/2 + 2/ dn1/d

Therefore, vg is given by vg = (d/d)/(d/d) = c/[n1 (dn1/d)] = c/Ng

where Ng is called the group refractive index or group index of the fibre.



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Power flow in a dielectric medium

If we take the cross product of E and H, we get a third vector acting in the

direction of propagation with the units of watt/m2 which is the power flow

per unit area. This vector is known as Poynting vectorS given by

S = E H

The average power is given by Sav = Re [(1/2) E H*].



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Planar waveguide

d/2

x = 0z

x

k1

k2

q

Combination of ray theory and wave theory (EM theory)

rktjEE uur

r

= exp0

The propagation constants k1 and k2 are vectors, which can be decomposed into

its x and z axis to give

q = (n12ko

2 2)1/2 p = (2 n22ko2 )1/2

where p and q are propagation constants in x-direction, while is the

propagation constant in the z-direction.



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When the critical angle exceeds in the slab, pbecomes imaginary, and can beexpressed in terms of a real attenuation

p = j2

when this equation is used with the wave equation and the boundary condition,

the solution obtained would be:

( )

( )

=

2exp

2exp

2cos

2expcos

),(dxforzjdxpqd

dxforzjqx

EzxEoy


The solution gives is the lowest order electric field TE0.

Ch2A Mode Theory Fundamentals

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