Propagation characteristics & modal analysis of different types of fibers

To observe the modal analysis and thepropagation characteristics of different typesof fibers

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Origin of optical fiber Introduction Principle of operation Physical parameters of optical fiber Types of fibers Different types of losses and dispersion Ray propagation Modal analysis Matlab programs Application

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Nature had found away to guide light.The light can beguided around anycomplex path.

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Fiber optics is awaveguide used totransmit television,voice and digitalsignals by light wavesover flexible hair likethreads of glass andplastic. It has evolvedinto a system of greatimportance and usedsince 1980’s.

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Greater bandwidth Small size and weight Electrical isolation Immunity to interference and crosstalk Signal security Low transmission loss Flexibility Reliability Ease of maintenance Low cost

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Optical fiber communication system

InputSignal

Coder orConverter

LightSource

Source-to-FiberInterface

Fiber-to-lightInterface

LightDetector

Amplifier/ShaperDecoder

Output

Fiber-optic Cable

Transmitter

Receiver

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The thin glass center of the fiber where the light travels is called the “core”.

The outer optical material surrounding the core that reflects the light back into the core is called the “cladding”.

In order to protect the optical surface from moisture and damage, it is coated with a layer of buffer coating. .

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The operation of optical fiber is based on the principle of total internal reflection of light. Total internal reflection is an optical phenomenon where a ray of light travelling from optically denser medium to lighter medium is reflected back into the denser medium at the interface when the angle of incidance at the denser medium is greater than critical angle.

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If θ < θc, as with the red ray in the above figure, the ray will split. Some of the ray will reflect off the boundary, and some will refract as it passes through.

If θ > θc, as with the blue ray, the entire ray reflects from the boundary. None passes through. This is called total internal reflection.

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ACCEPTANCE ANGLE NUMERICAL APERTURE V-PARAMETER

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ACCEPTANCE ANGLE-the largest incident angle ray that can be

coupled into a guided ray within the fiber .

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NUMERICAL APERTURE The light gathering power of an optical fiber

is given by NA.

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V-PARAMETER It is an important parameter that governs

the number of modes

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ADVANTAGES1.Minimum dispersion2.Less attenuation3.Larger bandwidth

DISADVANTAGES1.Difficult to couple light2.Expensive

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ADVANTAGES1.Larger core diameter

DISADVANTAGES1.Suffer from intermodal dispersion

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Have constant refractive indexRefractive index profile n(r)= n1 when r<a (core)

n2 when r≥a (cladding) where n1= refractive index of core

n2= refractive index of cladding a=core radius

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Refractive index continuously decreases from the core radially outwards towards the cladding

Refractive index profilen(r)=n1(1-2Δ(r/a)α )1/2 when r<a (core) n1(1-2Δ)1/2 =n2 when r≥a (cladding)

where n1=refractive index of coren2=refractive index of claddingΔ=relative refractive index difference a=core radius

α=profile parameter

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1.MULTIMODE STEP INDEX FIBER Core diameter=50 to 400µmCladding diameter=125 to 500µmBuffer jacket diameter=250 to 1000µmNumerical aperture=0.16 to 0.52.MULTIMODE GRADED INDEX FIBER Core diameter=30 to 100µmCladding diameter=100 to 150µmBuffer jacket diameter=250 to 1000µmNumerical aperture=0.2 to 0.3 3.SINGLE MODE FIBER Core diameter=5 to 10µm, typically around 8.5µmCladding diameter=generally 125µmBuffer jacket diameter=250 to 1000µmNumerical aperture=0.08 to 0.15 usually around 0.10

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ATTENUATION It is the loss of optical power as light travels

down a fiber. It is measured in decibels (dB/km). Over a set distance, a fiber with a lower attenuation will allow more power to reach its receiver than a fiber with higher attenuation.

ABSORPTION It is a loss mechanism related to the material

composition and the fabrication process for the fiber, which results in the dissipation of some of the transmitted optical power as heat in the waveguide.

SCATTERING Scattering, another significant aspect of

attenuation, occurs when atoms or other particles within the fiber spread the light.

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the "spreading" of a light pulse as it travels down a fiber

1. INTERMODAL- results from propagation delay differences between modes with in a multimode fiber.

2. INTRAMODAL- occur in all types of optical fiber and result from the finite spectral line width of the optical source.

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Two types of rays propagating along the fiberare-

MERIDIONAL RAYS-pass through the axis of the optical fiber

SKEW RAYS- travel through an optical fiber without passing through its axis

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Skew rays

Meridional rays

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IN STEP INDEX & GRADED INDEX FIBER

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For step index type fiber let us take the guide layer(core) ofrefractive index n1 and the cladding of refractive index n2 where n1

and n2 are constants n1>n2. For TE modes the partial derivativeinvolving Ey is written as d²Ey - β²Ey + μ0ε0ω²n²(x)Ey=0 ………………………….eqn(1)dx² or, d²Ey + (k²n²(x)- β²)Ey=0 …………………………….eqn(2) dx² where μ0ω0=1/c² and ω/c=k , k is the free space wavenumber n(x)= n1 for |x| < a

n2 for |x| > awhere a is the maximum radial distance of the core

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d²Ey + (k²n1²- β²)Ey=0 (|x| < a)………………………………eqn(3)

dx² d²Ey + (k²n2²- β²)Ey=0 (|x| > a)……………………………eqn(4)

dx² Let us put, u² = k²n1² - β² = β1²- β²

w² = β² - k²n2² = β² - β2²

Therefore d²Ey + u²Ey=0 (|x| < a)…………………………………………eqn(5)

dx²d²Ey + w²Ey=0 (|x| > a)………………………………………..eqn(6)

dx² For the wave to be guided through the layerΒ1²(=k²n1²)> β²> β2²(=k²n2²)

V-Parameter is then derived asV={(ua)²+(wa)²}½ = (2πa/λ)*(n1²-n2²)½……………………….eqn(7)

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Therefore normalized propagation constant bis given as follows B=(β²-β2²)/(β1²-β2²)=( β²-n2²k²)/(n1²k²-n2²k²)=w²/v²=1-u²/v²……….eqn(8)

Equation for an electric field in an isotropic, linear, non conducting, nonmagnetic but inhomogeneous medium is given by ▼²E + ▼{(1/εr) ▼( εr).E} - μoεoεr∂²E = 0…………………………………eqn(9)

∂t²▼²H +(1/ εr) {▼(εr) × (▼× H)} - μoεoεr∂²H = 0…………………………eqn(10)

∂t²

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Since n may depend on the transverse coordinates (r,φ) and the wave ispropagating along the z direction, we may write the solution as Ψ(r,φ,z,t)=ψ(r,φ)ei(ωt-βz)

Putting the value of ψ and εoμo=1/c² and ω/c=k

we get ∂²ψ + 1/r ∂ψ + 1/r² ∂²ψ + (n²k²-β²)ψ = 0………………………..eqn(11) ∂r² ∂r ∂φ² Applying variable separation methodΨ(r,φ)=R(r)Ф(φ)

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Using the equations of u and w the equations are modified to r²d²R + rdR + [u²r² – l²]R = 0, r ≤ a ……….eqn(12) dr² dr r²d²R + rdR - [w²r² + l²]R = 0, r ≤ a …………eqn(13) dr² dr Equations 1 and 2 are second order equations and have twoindependent solutions. The solutions of equation 1 are the Besselfunction of the first kind and the modified Bessel function of thefirst kind. The solutions of equation 2 are the Bessel function ofthe second kind and the modified Bessel function of the secondkind. Modified Bessel function of first kind has discontinuity at theorigin and the Bessel function of the second kind has anasymptotic form. So these are discarded.

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In a typical step index fiber the difference between the indices of refraction of the core

and cladding is very small i.e Δ<<1.this is called weakly guiding fiber approximation.

In this approximation the electromagnetic field patterns and the propagation constants of the mode pairs HEl+1,m and EHl-

1,m

are very similar.

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So mode groupings are done in the following manner(l,m)=(0,1) …….HE11

(2,1)…….. HE21, TE01, TM01

(1,2)………HE12 So LPlm mode corresponds to HEl+1,m

and EHl-1,m

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So for LP01 mode l=0 and m=1

So for LP01 mode only HE11 mode is passing Again for LP11 mode l=1 and m=1

So this corresponds to HE21 mode which again

corresponds to TE01 and TM01 modes.

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-25 -20 -15 -10 -5 0 5 10 15 20 25-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

radial distance(r) (um)

V-P

aram

eter

(V)

(rad

)V-Parameter vs radial distance from the axis of the fiber

According to formula V number=2*pi*a/λ(NA)

As V is directly proportional to the radius of the fiber

So the plot is a simple straight line passing through the origin

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0 5 10 15-0.5

0

0.5

1J(

u)

u

Plot of Bessel Functions indicating the allowed values of u for a lower order modes

For step index fiber with a constant refractive index core, the wave eqn is a bessel differential eqn.

In the core region the solution is a bessel function of first kind denoted by Jl.

A graph of these gradually damped oscillatory functions is shown in above figure

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1 1.5 2 2.50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

V-Parameter(V)(rad)

w/a

Normalized spot size (w/a) as a function of V

For step index fiber, mode field radius w is expressed as w=a[0.65+1.619/(V^3/2)+2.879/(V^6)]

where a=core radius Plot of normalized spot size (w/a) as a

function of V is done here It is clear from this plot that as normalized

spot size increases V becomes smaller or λ becomes larger.

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As wavelength increases the modal field is less well confined within the core

For single mode fibers the cut off wavelength is not too far from the wavelength for which the fiber is designed.

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0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V-Parameter(V) (rad)

norm

aliz

ed p

ropa

gatio

n co

nsta

nt(b

)

Variation of the normalized propagation parameter b for TE modes with V for a planar waveguide

Variation of V with b The cut offs of various modes areJl(Vc)=0 for l=0 modes

J0(Vc)=0 for l=1 modes

Jl-1(Vc)=0 for l>1 modes

where (Vc)= cut off V value=normalized cut off

Frequency So V value corresponds to zeros of bessel

functions So first V value and then b is calculated

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-80 -60 -40 -20 0 20 40 60 800

1

2

3

4

5

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radial distance from the axis of the fiber(r)

refr

activ

e in

dex(

n(r)

)

Refractive index profile of a single mode fiber

Commonly used single mode fibers have more or less step index profiles.

Design is optimized for operation at the minimum dispersion 1.3µm region.

So either matched cladding or depressed cladding fibers are formed

Figure shown above is matched cladding fiber Has a uniform refractive index in the cladding

which is slightly less than that of core.

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-80 -60 -40 -20 0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


refr

active index(n

(r))

Refractive index profile of a double clad fiber

Dispersion flattened profile Depressed cladding profile Good for WDM of optical fibers Design is highly sensitive to bend losses

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-80 -60 -40 -20 0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


refr

active index(n

(r))

Refractive index profile of a quadruple clad fiber

Depressed cladding profile Light loss through the bending can be

retrapped by further introducing regions of raised index into the structure.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

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normalized propagation constant

transcedenta

l equation

plot of transcendental equation with normalized propagation constant with V=2 and l=0

Variation of V(1-b)½J1[V(1-b)½]/J0[V(1-b)½] with b ( the propagation constant)

It is a universal curve showing the dependence of b ( and therefore u and w ) on V

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15

-10

-5

0

5

10

15


tran

sced

enta

l equ

atio

n

plot of transcendental equation with normalized propagation constant with V=6.5 and l=0

By studying the zeros of the bessel functions it is found b always lies between 0 and 1

This implies there will be only a finite no of guided modes

This is shown by a simple examplef(b) = V(1-b)½J1[V(1-b)½]/J0[V(1-b)½]

f(b)=0 when V(1-b)½=0,3.8317,7.0156f(b)=infinity when V(1-b)½=2.4048,5.5201For V=6.5 , b= 1,0.625 in case of f(b)=0and b=0.8631,0.2788 in case of f(b)=infinity

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By knowing the positions of zeros and infinities and f(b)=+ve in the vicinity of b=1, a qualitative plot can be made and the no of modes propagating and also the approximate value of b can be estimated

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.8

1

1.2

1.4

1.6

1.8

2


tran

sced

enta

l equ

atio

n

plot of transcendental equation with normalized propagation constant with V=2 and l=1

Variation of V(1-b)½Jl-1[V(1-b)½]/Jl[V(1-b)½] for l=1 and V=2

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15

-10

-5

0

5

10

15


tran

scede

ntal

equ

atio

n

plot of transcendental equation with normalized propagation constant with V=6.5 and l>=1

l = 1

l = 2

l = 3l = 4

l = 5

Variation of V(1-b)½Jl-1[V(1-b)½]/Jl[V(1-b)½] for V=6.5 corresponding to l=1,2,3,4,5 with the propagation constant i.e b

We can find the no of guided modes with the help of this plot

Approximate value of b can also be estimated

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01

23

4

-2

-1

0

1

24

4.5

5

5.5

6

phase

modal analysis of LP01 mode

radial distance

ampl

itude

This shows the field distribution of LP01 mode.

Plot is done by surfl command. The surfl function displays a shaded surface

based on a combination of ambient, diffuse, and specular lighting models.

Surfl create three-dimensional shaded surfaces using the default direction for the light source and the default lighting coefficients for the shading model.

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In this case amplitude, radius and phase are taken as the three components of a surface

It is been found that amplitude of light is maximum in the core region and it starts decreasing radially outwards towards the cladding.

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0 0.5 1 1.5 2 2.5 3 3.5-1.5

-1

-0.5

0

0.5

1

1.5

phase

modal analysis of LP01mode

radi

al d

ista

nce

This plot is done by surface command. Surface is the low-level function for creating

surface graphics objects. In this plot also it is found that as light is

been guided through the core so the middle portion is brightened and while moving towards the cladding the intensity of light starts decreasing

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0

1

2

3

4

-1.5-1

-0.50

0.51

1.54

4.5

5

5.5

6

phase

modal analysis of LP01mode

radial distance

ampl

itude

This is the different orientation of the above plot.

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-1.5 -1 -0.5 0 0.5 1 1.54.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

radial distance

ampl

itude


This is a 2D plot using the simple command plot.

The same explanation also prevails here.

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01

23

4

-2

-1

0

1

20

1

2

3

4

phase


radial distance

ampl

itude

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02

4-1.5-1-0.500.511.5

0

0.5

1

1.5

2

2.5

3

3.5

phaseradial distance

modal analysis of LP11 modeam

plitu

de

This is the different orientation of the above plot.

For LP11 mode m=1 i.e there is one zero in the transverse field pattern.

The zero is found on the axis and so the electric field is maximum on the left side of the zero and minimum on the right side of the zero. So the left side of the plot is bright and the right side is dark.

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Used as a medium for telecommunication and networking because it is flexible and can be bundled as cables.

It is especially advantageous for long-distance communications, because light propagates through the fiber with little attenuation compared to electrical cables.

A single fiber can carry much more data than a single electrical cable.

So fiber can easily replace cables.

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Fibers have many uses in remote sensing to measure strain, temperature and pressure.

Fibers are widely used in illumination applications. Optical fiber illumination is also used for decorative applications.

Optical fiber is also used in imaging optics. A coherent bundle of fibers is used, thin

imaging device called an endoscope, which is used to view objects through a small hole.

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1.Voice communication2. Video communication3. Data transfer4. Internet5. Sensor systems

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Microstructured holy fiber is a new class of optical fiber based on the properties of photonic crystals. Because of its ability to confine light in hollow cores or with confinement characteristics not possible in conventional optical fiber.It is now finding applications in fiber-optic communications, fiber lasers, nonlinear devices, high-power transmission, highly sensitive gas sensors, and other areas.

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We have studied Relationship between different parameters

of optical fiber Refractive index of different clad fibers Modal analysis of LP01 and LP11 mode

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With the help of the programs & numerical analysis that we have done in this project the modal analysis of micro structured holy fiber which is not yet developed can be derived. The amount of light passing through the modes & the coupling taking place in different modes can also be analyzed.

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SONALI DAS(22) SAUGATA DAS(21) RAHUL CHATTOPADHYAY(14) SHUBHABRATA MUKHERJEE(43)

PROJECT GUIDE

‣ KAKALI DAS

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Propagation characteristics & modal analysis of different types of fibers

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Transcript of Propagation characteristics & modal analysis of different types of fibers