Guided Propagation Along the Optical Fiber

Xavier FernandoRyerson University

The Nature of Light• Quantum Theory – Light consists of

small particles (photons)• Wave Theory – Light travels as a

transverse electromagnetic wave• Ray Theory – Light travels along a

straight line and obeys laws of geometrical optics. Ray theory is valid when the objects are much larger than the wavelength (multimode fibers)

Refraction and reflection

Snell’s Law: n1 Sin Φ1 = n2 Sin Φ2

Critical Angle:Sin Φc=n2/n1

Step Index Fiber

Core and Cladding are glass with appropriate optical properties while buffer is plastic for mechanical

protection

n1 n2

n1>n2

Step Index Fiber

n

y

n2 n1

Cladding

Core z

y

rφ

Fiber axis

The step index optical fiber. The central region, the core, has greater refractivindex than the outer region, the cladding. The fiber has cylindrical symmetryuse the coordinates r, φ, z to represent any point in the fiber. Cladding isnormally much thicker than shown.

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

Meridian Ray Representation

1

22

1

22

21 12 n

nn

nn−≈

−=∆

Total Internal Reflection

Cladding

Coreα < αm ax

AB

θ < θc

A

B

θ > θc

α > αm ax

n0 n1

n2

Lost

Propagates

Maximum acceptance angleαmax is that which just givestotal internal reflection at thecore-cladding interface, i.e.when α = αmax then θ = θc.Rays with α > αmax (e.g. rayB) become refracted andpenetrate the cladding and areventually lost.

Fiber axis

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

Graded Index Fiber

nb

nc

O O'Ray 1

A

B'

B

θAθB

θB' Ray 2

M

θB' c/nb

c/na12

B''na

a

b

c We can visualize a graded indefiber by imagining a stratifiedmedium with the layers of refrindices na > nb > nc ... Consider twclose rays 1 and 2 launched froOat the same time but with slighdifferent launching angles. Rayjust suffers total internal reflecRay 2 becomes refracted at B andreflected at B'.

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

Single Mode Step Index Fiber

Protective polymerinc coating

Buffer tube: d = 1mm

Cladding: d = 125 - 150 µm

Core: d = 8 - 10 µmn

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)

Fiber Key Parameters

Comparison of fiber structures

Fiber Key Parameters

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

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

Step and Graded Index Fibers

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)

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

Total Internal Reflection

Fiber axis

12

34

5

Skew ray1

3

2

4

5

Fiber axis

1

2

3Meridional ray

1, 3

2

(a) A meridionaray alwayscrosses the fibeaxis.

(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

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

Skew Rays

Skew rays

Skew rays circulate around the core and increase the dispersion

Polarizations of fundamental mode

Two polarization states exist in the fundamental mode in a single mode fiber

Polarization Mode Dispersion (PMD)

Each polarization state has a different velocity PMD

Major Dispersions in Fiber• Modal Dispersion: Different modes travel at

different velocities, exist only in multimodal conditions

• Waveguide Dispersion: Signal in the cladding travel with a different velocity than the signal in the core, significant in single mode conditions

• Material Dispersion: Refractive index n is a function of wavelength, exists in all fibers, function of the source line width

Effects of Dispersion and Attenuation

Dispersion for Digital Signals

t0

Emitter

Very shortlight pulses

Input Output

Fiber

PhotodetectorDigital signal

Information Information

t0

~2² τ1/2

T

t

Output IntensityInput Intensity² τ1/2

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

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

Low order modeHigh order mode

Cladding

Core

Light pulse

t0 t

Spread, ∆τ

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.

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

Modal Dispersion

n2

Light

n2

n1

y

E(y)

E(y,z,t) = E(y)cos(ωt – β0z)

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 θ. It is often referred to as theglazing incidence ray. It has the highest phase velocity along the guide.

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

Field Distribution in the Fiber

Higher order modes Larger MFDy

E(y)m = 0 m = 1 m = 2

Cladding

Cladding

Core 2an1

n2

n2

The electric field patterns of the first three modes (m = 0, 1, 2)traveling wave along the guide. Notice different extents of fiepenetration into the cladding.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Mode-field Diameter (2W0)

In a Single Mode Fiber,

)/exp()( 20

20 wrErE −=

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

Typically Wo > a

Power in the cladding

Lower order modes have higher power in the cladding.

y

E(y)

Cladding

Cladding

Core

λ2 > λ1λ1 > λc

ω2 < ω1ω1 < ωcut-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.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Higher the Wavelength More the Evanescent Field

Mode-field diameter Vs wavelength

E

r

E01

Core

Cladding

The electric field distribution of the fundamental modin 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

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

Cladding Power Vs Normalized Frequency

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.

0

0.5

1

1.5

0 1 2 3V - 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)

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

τt

Spread, ² τ

t0

λ

Spectrum, ² λ

λ1λ2λo

Intensity Intensity Intensity

Cladding

CoreEmitterVery shortlight pulse

vg(λ2)vg(λ1)

Input

Output

All excitation sources are inherently non-monochromatic and emit within aspectrum, ² λ, 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.

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

Material Dispersion

ZeroDispersionWavelength

Modifying Chromatic Dispersion

Chromatic Dispersion = Material dispersion + Waveguide dispersion• Material dispersion depends on the material

properties and difficult to alter• Waveguide dispersion can be altered by

changing the fiber refractive index profile– 1300 nm optimized– Dispersion Shifting (to 1550 nm)– Dispersion Flattening (from 1300 to 1550 nm)

Different Index Profiles

1300 nm optimized

Dispersion Shifted

Different Index Profiles

Dispersion Flattened

Large area dispersion shifted Large area dispersion flattened

Different dispersionprofiles

Dispersion Shifting/Flattening

20

-10

-20

-30

10

1.1 1.2 1.3 1.4 1.5 1.6 1.7

0

30

λ (µm)

Dm

Dw

Dch = Dm + Dw

λ1

Dispersion coefficient (ps km -1 nm-1)

λ2

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 λ1 and λ2.

Zero Dispersion Wavelength

0

1.2 1.3 1.4 1.5 1.61.1-30

20

30

10

-20

-10

λ (µ m)

Dm

Dm + D w

D wλ0

Dispersion coefficient (ps km -1 nm -1 )

M ateria l d i sp ersio n co effi c i en t (D m ) fo r th e co re m ateria l (t ak en asS iO 2), w av eg u id e d i sp ersio n co effi c ien t (D w ) (a = 4 .2 µ m ) an d th eto ta l o r ch ro m at ic d i sp ersio n co effi c ien t D ch (= D m + D w ) as afu n ct io n o f free sp ace w av elen g th , λ .

Material and waveguide dispersion coefficients in anoptical fiber with a core SiO 2-13.5%GeO 2 for a = 2.5to 4 µm.

0

–10

10

20

1.2 1.3 1.4 1.5 1.6–20

λ (µm )

D m

D w

SiO 2-13.5% GeO 2

2.53.03.54.0a (µm )

Dispersion coefficient (ps km -1 nm -1)

Total Dispersion

For Single Mode Fibers:

For Multi Mode Fibers:

Group Velocity Dispersion

If PMD is negligible

Dispersion & Attenuation Summary

t0

Pi = Input light power

Emitter

OpticalInput

OpticalOutput

Fiber

PhotodetectorSinusoidal signal

Sinusoidal electrical signalt

t0f

1 kHz 1 MHz 1 GHz

Po / Pi

fop

0.10.05

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

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)Fiber Optic Link is a Low Pass Filter for Analog Signals

Attenuation Vs Frequency

Attenuation in FiberAttenuation 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 −=α

All Wave Fiber for DWDM

Lowest attenuation occurs at 1550 nm for Silica

Att

enua

tion

char

acte

rist

ics

Escaping wave

θ θ

θ′ < θ

θθ > θc θ′

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 θ′ 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

Power loss in a curved fiber

Power in the evanescent field evaporates first

Bending-induced attenuation

Bending effects on loss Vs MFD

Micro-bending losses

Fiber Production

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