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Mode Theory for Optical Fiber

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Transverse Electric (TE) Modes

Electric field is perpendicular to the direction of propagation.

Most of the magnetic field is also perpendicular to the z-direction but a small z-component exists.

The “ray” path is meridional.

It is not circular or skewed.

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Transverse Magnetic (TM) Modes

Magnetic field is perpendicular to the direction of propagation.

Most of the electric field is also perpendicular to the z-direction but a small z-component exists.

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Transverse Electromagnetic (TEM) Modes

In the TEM mode both the electric and magnetic fields are perpendicular to the direction of propagation i.e. z.

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Helical (Hydrid) Modes (HE and EH)

In a fibre, many modes also travel in a helical path of some kind.

In this case components of both magnetic and electric fields are in the direction of propagation. Ez ≠ 0 and Hz ≠ 0.

These modes are designated as either HE or EH depending on which field contributes the most to the z-direction.

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Skew Rays in Fiber

Skew rays do not pass through the fibre centre axis. Skew rays greatly outnumber meridional rays. Skew rays follow a helical path within the fibre.

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Skew Rays in Fiber

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Acceptance Angle For Skew Rays

S

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Wave equation for optical waveguide are given as

2

22

t

2

2

22 1

tv p

It is the Laplacian operater. In cylindrical coordinate system it is given as:

2

2

2

2

22

22 11

zrrrrr

Here ψ represents Electric field (E) or Magnetic field (H)

)()( ztjjlzz eer

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If we solve the wave equation of optical waveguide for t, z and ϕ we get a differential equation:

01

2

2222

12

2

zzz

r

lkn

rrr

The above equation is a Bessel differential equation, for core region the solutions are Bessel function denoted by Jl.

022221

22

22

zzz lknr

rr

rr

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Bessel functions are a series of solutions to a second order differential equation

Applied to the situation with cylindrical symmetry. Solutions of Bessel function is

Jl(x) is the Bessel function of first kind and order l. A graph of this with x is shown in the next slide.

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The field is finite at r = 0 and it is represented by zero order Bessel function J0

As r increases and goes towards infinity the field vanishes and in the cladding this solution is given by another function called Modified Bessel functions denoted by Kl.

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Graph of modified Bessel function Kl(r) against r for l=0,1

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Cylindrical Fiber Modes

As with dielectric waveguide TE and TM modes are common.

The cylindrical fiber is bound in two directions hence two integers l and m are necessary to specify the modes.

For cylindrical waveguides we refer to TElm, TMlm, HElm and EHlm modes.

The EH and HE modes are due to nonzero Ez and Hz components.

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OF support weakly guiding approximation, where Relative RI difference (∆) is very small as compared to unity.

∆<< 1, generally ∆ is taken less then 0.03 i.e. 3%. In fiber communication actual field distribution is the

combination of modes rather than any one mode. Thus TE, TM, HE and EH modes can all be explained

using only a single set of Linearly Polarized modes. LP modes are not the exact fiber modes but they are

resultant modes due to approximation.

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For these guiding structures the mode theory gives dominant transverse field components, i.e. the approximate solutions for the full set of HE, EH, TE and TM modes, given by two linearly polarized components called LP modes.

LP modes also use subscripts l and m.

2l field minima exist around circumference of core and m field maxima along radius vector.

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Correspondence Between LP Modes And Traditional Exact Modes

Linearly Polarized Mode Exact Modes LP01 HE11 LP11 HE21,TE01, TM01 LP21 HE31, EH11 LP02 HE12

LP31 HE41, EH21

LP12 HE22, TE02, TM02

LP0m HE1m

LPlm HE2m, TE0m, TM0m

LPlm(l >1) HEl+1 m, EHl-1 m

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Energy Distribution of Some LP Modes in Fiber

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Vc is different for different modes.

First zero crossing of J1 occurs at V = 0 and this corresponds to cutoff for LP01 mode.

The first zero crossing for J0 is when V = 2.405 giving Vc for LP11 mode.

Second zero crossing for J1 occurs at V = 3.83 giving Vc

for LP02 of 3.83

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Hence we can have fibers manufactured with a particular V no. for limiting certain modes.

It may be further noted that cutoff value of V i.e. Vc occurs when β = n2k correspond to b = 0

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U and W are equal to the eigen-values in the core and cladding respectively defined as

U = a(n12k2 – β2)1/2

W = a(β2 – n22k2)1/2

A very useful relation using U and W can be given byV = (U2 + W2)1/2

V = ka(n12 – n2

2)1/2

V is called the normalized frequency or V number. It is a dimensionless parameter. Expression relates core radius, relative reflective index

difference (∆) and wavelength.

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Another important parameter can also define is the normalized propagation constant b for a fiber as

b = 1 – (U2/V2)

b = [(β/k)2 – n22] / (n1

2 – n22)

b = [(β/k)2 – n22] / 2n1

2∆

The limit of β is between n1k and n2k but b lies between 0 - 1

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b-V Curve

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Unguided or radiating modes have frequencies below cutoff where β < n2k and W is imaginary or zero. These modes are also called leaky modes.

As β moves toward n2k less and less power is propagated in cladding and when it becomes β = n1k all power is confined to core.

Range of β signifies the guided modes in the fiber.

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Now we have b and β as function of V, propagation characteristics of modes and their dependence on frequency and fiber geometry can be obtained.

When β = n2k, then mode is no longer properly guided.

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Mode Coupling Due to waveguide perturbation such:

Deviation of fiber axis from straigthness. Variation in the core diameter. Irregularities at the core-cladding interface. Refractive index variations.

It may change the propagation characteristics of fiber. Result in coupling of energy traveling in one mode to

another. Hence individual mode do not propagate throughout

the fiber.

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Mode Field Diameter

Modes in Fiber are defined on the EM field distribution in radial direction.

MFD is important in SM operation. Normally modes are not confined in core only. There is field penetration in the cladding depending

on wave length. The diameter of the fiber region in which propagation

of any mode occurs is called as MFD.

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The field distribution can be well approximated by Gaussian Distribution.

MDF is equal to the width in which Gaussian field strength is more then or equal to the 1/e times of maximum value.

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