The Optical Fiber and Light Wave Propagation Xavier Fernando Ryerson Comm. Lab.

The Optical Fiber and Light Wave Propagation

Xavier Fernando

Ryerson Comm. Lab

The Optical Fiber• Fiber optic cable functions as a ”light

guide,” guiding the light from one end to the other end.

• Fiber categories based on propagation:– Single Mode Fiber (SMF)– Multimode Fiber (MMF)

• Categories based on refractive index profile– Step Index Fiber (SIF)– Graded Index Fiber (GIF)

Step Index Fiber• Uniform ref. index of n1 (1.44 < n1 < 1.46) within

the core and a lower ref. index n2 in the cladding.

• The core and cladding radii are a and b. Typically 2a/2b are 8/125, 50/125, 62.5/125, 85/125, or 100/140 µm.

• SIF is generally made by doping high-purity fused silica glass (SiO2) with different concentrations of materials like titanium, germanium, or boron.

n1 n2

n1>n2

Different Light Wave Theories

• Different theories explain light behaviour • We will first use ray theory to understand

light propagation in multimode fibres• Then use electromagnetic wave theory to

understand propagation in single mode fibres

• Quantum theory is useful to learn photo detection and emission phenomena

Refraction and Reflection

Snell’s Law: n1 Sin Φ1 = n2 Sin Φ2

When Φ2 = 90,

Φ1 = Φc is the

Critical Angle

Φc=Sin-1(n2/n1 )

Step Index Multimode Fiber

1

221

22

21 12 n

n

n

nn

Fractional refractive-index

profile

Ray description of different fibers

Single Mode Step Index Fiber

Protective polymerinc coating

Buffer tube: d = 1mm

Cladding: d = 125 - 150 m

Core: d = 8 - 10 mn

r

The cross section of a typical single-mode fiber with a tight buffertube. (d = diameter)

n1

n2

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Only one propagation mode is allowed in a given wavelength. This is achieved by very small core diameter (8-10 µm)SMF offers highest bit rate, most widely used in telecom

Step Index Multimode Fiber

• Guided light propagation can be explained by ray optics

• When the incident angle is smaller the acceptance angle, light will propagate via TIR

• Large number of modes possible• Each mode travels at a different velocity

Modal Dispersion

• Used in short links, mostly with LED sources

Graded Index Multimode Fiber

• Core refractive index gradually changes towards the cladding

• The light ray gradually bends and the TIR happens at different points

• The rays that travel longer distance also travel faster

• Offer less modal dispersion compared to Step Index MMF

Refractive Index Profile of Step and Graded Index Fibers

a b

n1

n2n2

n1

a b

n = n =

Ste

p

Gra

ded

Step and Graded Index Fibers

n1

n2

21

3

nO

n1

21

3

n

n2

OO' O''

n2

(a) Multimode stepindex fiber. Ray pathsare different so thatrays arrive at differenttimes.

(b) Graded index fiber.Ray paths are differentbut so are the velocitiesalong the paths so thatall the rays arrive at thesame time.

23


Total Internal Reflectionin Graded Index Fiber

n decreases step by step from one layerto next upper layer; very thin layers.

Continuous decrease in n gives a raypath changing continuously.

TIR TIR

(a) A ray in thinly stratifed medium becomes refracted as it passes from onelayer to the next upper layer with lower n and eventually its angle satisfies TIR.(b) In a medium where n decreases continuously the path of the ray bendscontinuously.

(a) (b)


Total Internal Reflectionin Graded Index Fiber - II

nb

nc

O O'Ray 1

A

B'

B

AB

B' Ray 2

M

B' c/nb

c/na12

B''

na

a

b

c We can visualize a graded indexfiber by imagining a stratifiedmedium with the layers of refractiveindices na > nb > nc ... Consider twoclose rays 1 and 2 launched from Oat the same time but with slightlydifferent launching angles. Ray 1just suffers total internal reflection.Ray 2 becomes refracted at B andreflected at B'.


Skew Rays

Fiber axis

12

34

5

Skew ray1

3

2

4

5

Fiber axis

1

2

3Meridional ray

1, 3

2

(a) A meridionalray alwayscrosses the fiberaxis.

(b) A skew raydoes not haveto cross thefiber axis. Itzigzags aroundthe fiber axis.

Illustration of the difference between a meridional ray and a skew ray.Numbers represent reflections of the ray.

Along the fiber

Ray path projectedon to a plane normalto fiber axis

Ray path along the fiber


Maxwell’s Equations

……..(1) (Faraday’s Law)

E: Electric Field

……….(2) (Maxwell’s Faraday equation)

H: Magnetic Field

……….(3) (Gauss Law)

……….(4) (Gauss Law for magnetism)

Taking the curl of (1) and using and

The parameter ε is permittivity and μ is permeability.

…….(5)

In a linear isotropic dielectric material with no currents and free of charges,

Maxwell’s Equations

But from the vector identity

……(6)

Using (5) and (3), …….(7)

Similarly taking the curl of (2), it can be shown

………(8)

(7) and (8) are standard wave equations. Note the Laplacian operation is,

2

2

2

2

22 11

z

EE

rr

Er

rrE

1

Note pv

Maxwell’s Equation

• Electrical and magnetic vectors in cylindrical coordinates are give by,

.…..(9)

……(10)

• Substituting (9) and (10) in Maxwell’s curl equations

….(11)

….(12)

….(13)

Maxwell’s Equation• Also

----------(14)

----------(15)

----------(16)

• By eliminating variables, above can be rewritten such that when Ez and Hz are known, the remaining transverse components Er , Eφ, Hr , Hφ, can be determined from (17) to (20).

Maxwell’s Equation …………..(17)

…………..(18)

…..........(19)

.………… (20)

Substituting (19) and (20) into (16) results in ….…(21)

…….(22)

Electric and Magnetic ModesNote (21) and (22) each contain either Ez or Hz only. This may

imply Ez and Hz are uncoupled. However. Coupling between Ez and Hz is required by the boundary conditions.

If boundary conditions do not lead to coupling between field components, mode solution will imply either Ez =0 or Hz =0. This is what happens in metallic waveguides.When Ez =0, modes are called transverse electric or TE modes

When Hz =0, modes are called transverse magnetic or TM modes

However, in optical fiber hybrid modes also will exist (both Ez and Hz are nonzero). These modes are designated as HE or EH modes, depending on either H or E component is larger.

Wave Equations for Step Index Fibers

• Using separation of variables

………..(23) • The time and z-dependent factors are given by

………..(24)

• Circular symmetry requires, each field component must not change when Ø is increased by 2п. Thus

…………(25)• Where υ is an integer.• Therefore, (21) becomes

….(26)

Wave Equations for Step Index Fibers• Solving (26). For the fiber core region, the solution must

remain finite as r0, whereas in cladding, the solution must decay to zero as r∞

• Hence, the solutions are – In the core, (r < a),

Where, Jv is the Bessel function of first kind of order v

– In the cladding, (r > a),

Where, Kv is the modified Bessel functions of second kind

Bessel Functions First Kind Bessel Functions Second kind

Modified Bessel first kind Modified Bessel Second kind

http://upload.wikimedia.org/wikipedia/commons/5/5d/Bessel_Functions_(1st_Kind,_n=0,1,2).svg

http://upload.wikimedia.org/wikipedia/commons/6/66/Bessel_Functions_(2nd_Kind,_n=0,1,2).svg

http://upload.wikimedia.org/wikipedia/commons/f/f3/BesselI_Functions_(1st_Kind,_n=0,1,2,3).svg

http://upload.wikimedia.org/wikipedia/commons/b/b9/BesselK_Functions_(n=0,1,2,3).svg

Propagation Constant β

• From definition of modified Bessel function

• Since Kv(wr) must go to zero as r∞, w>0. This implies that

• A second condition can be deduced from behavior of Jv(ur). Inside core u is real for F1 to be real, thus,

• Hence, permissible range of β for bound solutions is

Meaning of u and w

• Both u and w describes guided wave variation in radial direction– u is known as guided wave radial direction phase

constant (Jn resembles sine function)

– w is known as guided wave radial direction decay constant (recall Kn resemble exponential function)

Inside the core, we can write, 221

22 kuq

Outside the core, we can write, 22

22 kw

V-Number (Normalized Frequency)

All but HE11 mode will cut off when b = 0.

Hence, for single mode condition,

405.2)(2

cVNAa

V

22

21

222

22

12222 2

)( nna

akkawuV

Define the V-Number (Normalized Frequency) as,

2

221

22

2

2

22 )/(

nn

nk

V

awb

Define the normalized propagation const b as,

0 2 4 61 3 5V

b

1

0

0.8

0.6

0.4

0.2

LP01

LP11

LP21

LP02

2.405

Normalized propagation constant b vs. V-numberfor a step index fiber for various LP modes.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

0

0.5

1

1.5

0 1 2 3

V - number

V[d2(Vb)/dV2]

[d2(Vb)/dV2] vs. V-number for a step index fiber (after W.A. Gambling etal., The Radio and Electronics Engineer , 51, 313, 1981)


Field Distribution in the SMF

n2

Light

n2

n1

y

E(y)

E(y,z,t ) = E(y)cos(t – 0z)

m = 0

Field of evanescent wave(exponential decay)

Field of guided wave

The electric field pattern of the lowest mode traveling wave along theguide. This mode has m = 0 and the lowest . It is often referred to as theglazing incidence ray. It has the highest phase velocity along the guide.


Mode-field Diameter (2W0)

In a Single Mode Fiber,

)/exp()( 20

20 wrErE

At r = wo, E(Wo)=Eo/e

Typically Wo > a

Cladding Power Vs Normalized Frequency

Vc = 2.4

Modes

Power in the cladding

Lower order modes have higher power in the cladding larger MFD

Higher the Wavelength More the Evanescent Field

y

E(y)

Cladding

Cladding

Core

2 > 11 > c

2 < 11 < cut-off

vg1

y

vg2 > vg1

The electric field of TE0 mode extends more into thecladding as the wavelength increases. As more of the fieldis carried by the cladding, the group velocity increases.


Light Intensity

E

r

E01

Core

Cladding

The electric field distribution of the fundamental modein the transverse plane to the fiber axis z. The lightintensity is greatest at the center of the fiber. Intensitypatterns in LP01, LP11 and LP21 modes.

(a) The electric fieldof the fundamentalmode

(b) The intensity inthe fundamentalmode LP01

(c) The intensityin LP11

(d) The intensityin LP21


Fiber Key Parameters

Fiber Key Parameters

Effects of Dispersion and Attenuation

Dispersion for Digital Signals

t0

Emitter

Very shortlight pulses

Input Output

Fiber

PhotodetectorDigital signal

Information Information

t0

~2² T

t

Output IntensityInput Intensity

²

An optical fiber link for transmitting digital information and the effect ofdispersion in the fiber on the output pulses.


Modal DispersionLow order modeHigh order mode

Cladding

Core

Light pulse

t0 t

Spread,

Broadenedlight pulse

IntensityIntensity

Axial

Schematic illustration of light propagation in a slab dielectric waveguide. Light pulseentering the waveguide breaks up into various modes which then propagate at differentgroup velocities down the guide. At the end of the guide, the modes combine toconstitute the output light pulse which is broader than the input light pulse.


Major Dispersions in Fiber

• Modal Dispersion: Different modes travel at different velocities, exist only in multimode fibers

• This was the major problem in first generation systems

• Modal dispersion was alleviated with single mode fiber– Still the problem was not fully solved

Dispersion in SMF• Material Dispersion: Since n is a function of

wavelength, different wavelengths travel at slightly different velocities. This exists in both multimode and single mode fibers.

• Waveguide Dispersion: Signal in the cladding travels with a different velocity than the signal in the core. This phenomenon is significant in single mode conditions.

Group Velocity (Chromatic) Dispersion = Material Disp. + Waveguide Disp.

t

Spread, ²

t0

Spectrum, ²

12o

Intensity Intensity Intensity

Cladding

CoreEmitter

Very shortlight pulse

vg(2)

vg(1)Input

Output

All excitation sources are inherently non-monochromatic and emit within aspectrum, ² , of wavelengths. Waves in the guide with different free spacewavelengths travel at different group velocities due to the wavelength dependenceof n1. The waves arrive at the end of the fiber at different times and hence result ina broadened output pulse.


Group Velocity Dispersion

Modifying Chromatic Dispersion

GVD = Material Disp. + Waveguide dispersion

• Material dispersion depends on the material properties and difficult to alter

• Waveguide dispersion depends on fiber dimensions and refractive index profile. These can be altered to get:– 1300 nm optimized fiber– Dispersion Shifted Fiber (DSF) – Dispersion Flattened Fiber (DFF)

Material and Waveguide Dispersions

0

1.2 1.3 1.4 1.5 1.61.1-30

20

30

10

-20

-10

(m)

Dm

Dm + Dw

Dw0

Dispersion coefficient (ps km-1 nm-1)

Material dispersion coefficient (Dm) for the core material (taken asSiO2), waveguide dispersion coefficient (Dw) (a = 4.2 m) and thetotal or chromatic dispersion coefficient Dch (= Dm + Dw) as afunction of free space wavelength,


Material and waveguide dispersion coefficients in anoptical fiber with a core SiO2-13.5%GeO2 for a = 2.5to 4 m.

0

–10

10

20

1.2 1.3 1.4 1.5 1.6–20

(m)

Dm

Dw

SiO2-13.5%GeO2

2.5

3.03.54.0a (m)

Dispersion coefficient (ps km-1 nm-1)


Different WG Dispersion Profiles

WGD is changed by adjusting fiber profile

Dispersion Shifting/Flattening

(Standard)

(Zero Disp. At 1550 nm)

(Low Dispersion throughout)

20

-10

-20

-30

10

1.1 1.2 1.3 1.4 1.5 1.6 1.7

0

30

(m)

Dm

Dw

Dch = D m + Dw

1

Dispersion coefficient (ps km -1 nm-1)

2

n

r

Thin layer of claddingwith a depressed index

Dispersion flattened fiber example. The material dispersion coefficient ( Dm) for thecore material and waveguide dispersion coefficient ( Dw) for the doubly clad fiberresult in a flattened small chromatic dispersion between 1 and 2.


Specialty Fibers with Different Index Profiles

1300 nm optimized

Dispersion Shifted

Specialty Fibers with Different Index Profiles

Dispersion Flattened

Large area dispersion shifted Large area dispersion flattened

Polarization Mode Dispersion• Since optical fiber has a single axis of

anisotropy, differently polarized light travels at slightly different velocity

• This results in Polarization Mode Dispersion• PMD is usually small, compared to GVD or

Modal dispersion • May become significant if all other dispersion

mechanisms are small

X and Y Polarizations

A Linear Polarized wave will always have two orthogonal components.

These can be called x and y polarization componentsEach component can be individually handled if

polarization sensitive components are used

Polarization Mode Dispersion (PMD)

Each polarization state has a different velocity PMD

Birefringence• Birefringence is the decomposition of a ray of light

into two rays types of (anisotropic) material• In optical fibers, birefringence can be understood

by assigning two different refractive indices nx and ny to the material for different polarizations.

• In optical fiber, birefringence happens due to the asymmetry in the fiber core and due to external stresses

• There are Hi-Bi, Low-Bi and polarization maintaining fibers.

Total Dispersion

For Multi Mode Fibers:

For Single Mode Fibers:

But Group Velocity Disp.

Hence,

(ΔTpol is usually negligible )

(Note for MMF ΔTGVD ~= ΔTmat

Permissible Bit Rate• As a rule of thumb the permissible total

dispersion can be up to 70% of the bit period. Therefore,

T

0.35 B :bits) Zero(Return to pulses width halfFor •

T

0.7 B :bits) ZeroReturn to(Non pulses width fullFor •

Total

Total

Disp. & Attenuation Summary

t0

Pi = Input light power

Emitter

OpticalInput

OpticalOutput

Fiber

PhotodetectorSinusoidal signal

Sinusoidal electrical signalt

t0

f1 kHz 1 MHz 1 GHz

Po / Pi

fop

0.1

0.05

f = Modulation frequency

An optical fiber link for transmitting analog signals and the effect of dispersion in thefiber on the bandwidth, fop.

Po = Output light power

Electrical signal (photocurrent)

fel

10.707

f1 kHz 1 MHz 1 GHz


Fiber Optic Link is a Low Pass Filter for Analog Signals

Attenuation Vs Frequency

Fiber attenuation does not depend on modulation frequency

Attenuation in Fiber

Attenuation Coefficient

• Silica has lowest attenuation at 1550 nm• Water molecules resonate and give high

attenuation around 1400 nm in standard fibers• Attenuation happens because:

– Absorption (extrinsic and intrinsic)– Scattering losses (Rayleigh, Raman and Brillouin…)– Bending losses (macro and micro bending)

dB/km dB)(dB)0(

z

zPP

All Wave Fiber for DWDM

Lowest attenuation occurs at 1550 nm for Silica

Att

enu

atio

n

char

acte

rist

ics

Escaping wave

c

Microbending

R

Cladding

Core

Field distribution

Sharp bends change the local waveguide geometry that can lead to wavesescaping. The zigzagging ray suddenly finds itself with an incidenceangle that gives rise to either a transmitted wave, or to a greatercladding penetration; the field reaches the outside medium and some lightenergy is lost.


Bending Loss

Note:Higher MFD Higher Bending Loss

Micro-bending losses

Fiber Production

The Fiber Cable

Preform feed

Furnace 2000°C

Thicknessmonitoring gauge

Take-up drum

Polymer coater

Ultraviolet light or furnacefor curing

Capstan

Schematic illustration of a fiber drawing tower.


The Optical Fiber and Light Wave Propagation Xavier Fernando Ryerson Comm. Lab.

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