chap2 dielectric waveguides and optical...

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Topic 3Dielectric Waveguides and

Optical Fibers

Kasap Chapter 2

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Contents (Kasap Chapter 2)1 Symmetric Planar Dielectric Slab Waveguide2 Modal and Waveguide Dispersion in the Planar Waveguide3 Step Index Fiber4 Numerical Aperture5 Dispersion in Single Mode Fibers6 Bit Rate, Dispersion, Electrical and Optical Bandwidth7 The Graded Index (GRIN) Optical Fiber8 Light Absorption and Scattering9 Attenuation in Optical Fibers10 Fiber Manufacture

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2-1 Symmetric Planar Dielectric Slab Waveguide

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Light

n2

A planar dielectric waveguide has a central rectangular region ofhigher refractive index n1 than the surrounding region which hasa refractive index n2. It is assumed that the waveguide isinfinitely wide and the central region is of thickness 2a. It isilluminated at one end by a monochromatic light source.

n2

n1 > n2

Light

Light Ligh

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

cladding

core

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n2

n2

d = 2a

k1

Light

A

B

C

E

n1

A light ray travelling in the guide must interfere constructively with itself topropagate successfully. Otherwise destructive interference will destroy thewave.


z

y

x

TIR at B & Ck1 (AB+BC) + phase change due to TIR = m(2)

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/2 111 nknk ==

)(m)BCAB(k)AC( 221 =+=

Waveguide Condition

For constructive interference, the phase difference between A and C must be a multiple of 2

(1)

Dividing (2) by 2 we obtain the waveguide condition

(3)

man mm =

cos)2(2 1

[ ] ( ) 2221 mcosdk = (2)

cosdBC = )2cos( BCAB =

cos2]1)1cos2[()2cos( 2 dBCBCBCBCAB =+=+=+

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n2

n2

z2a

y

A

1

2 1

B

A

B

C2 2/2

k1E

x

n1

Two arbitrary waves 1 and 2 that are initially in phase must remain in phaseafter reflections. Otherwise the two will interfere destructively and cancel eachother.


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Resolve the wavevector k1 into two propagation constants, and , along and perpendicular to the the guide axis z

mmmnk

sin2sin 11

==

mmmnk

cos2cos 11

==

mmmm yakCAkACk == cos)(2')( 111

)()( mmm maymy +==

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n2

z

ay

A

1

2

A

C

kE

x

y

ay

Guide center

2

Interference of waves such as 1 and 2 leads to a standing wave pattern along the y-direction which propagates along z.

?1999 S O K O l i (P ti H ll)

mmmm yakCAkACk == cos)(2')( 111)()( mmm ma

ymy +==

)cos(),,( 01 mmm yztEtzyE ++=

)cos(),,( 02 yztEtzyE mm =

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)cos(),,( 01 mmm yztEtzyE ++=

)cos(),,( 02 yztEtzyE mm =

)21cos()

21cos(2),,( 0 mmmm ztyEtzyE ++=

)cos()(2),,( ztyEtzyE mm =

Traveling wave along zStanding wave along y

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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.

?1999 S O K O t l t i (P ti H ll)

)21cos()

21cos(2),,( 0 mmmm ztyEtzyE ++=

)cos()(2),,( ztyEtzyE mm =

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y

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 fieldpenetration into the cladding.?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

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mode of propagation mode number high order mode

m smaller, more bouncing, more penetration lowest order mode (m=0) : fundamental mode

m ~90 deg., travels axially

Low 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.


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cm sinsin >

Single and Multimode Waveguides

Vm 2

2122

21

2 /)nn(aV =V-number

= V-parameter= normalized thickness= normalized frequency

For given , V depends on the waveguide parameters a, n1, n2.

TIR conditionFrom (3)

mode number

V that makes m=0 single mode

V< /2, m=0 is the only possibility and only the fundamental mode (m=0) Propagates along the dielectric slab waveguide, which is then termed single mode planar waveguide.At c that leads to V= /2: cutoff wavelength

man mm =

cos)2(2 1

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E

By

Bz

z

y

O

B

E// Ey

Ez

(b) TM mode(a) TE mode

B//

x (into paper)

Possible modes can be classified in terms of (a) transelectric field (TE)and (b) transmagnetic field (TM). Plane of incidence is the paper.?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

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TE and TM modesTEm modes, transverse electric field modes

TMm modes,transverse magnetic field modes

The phase change that accompanies TIR depends on the polarization and is different forHowever for the waveguide condition and cutoff condition can be identical for both TE and TM modes.

xEEE x==

zEyEEE zy// +==

//EE and

121

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Example 2.1.1: Waveguide modes

Consider a planar dielectric guide with a core thickness 20m, n1 = 1.455, n2 = 1.440, light wavelength of 900nm. Given the waveguide condition in Eq(3) and the expression for in TIR for the TE mode,

man mm =

cos)2(2 1

using a graphical solution, find angles m for all the modes. What is your conclusion?

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10

5

082 84 86 88 90

m

f(m)

m = 0, evenm = 1, odd

89.17

88.34

87.5286.68

c

tan(ak1cosm m/2)

Modes in a planar dielectric waveguide can be determined byplotting the LHS and the RHS of eq. (11).?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

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Example 2.1.2: V-number and the number of modes

Using Eq. (9), mode number =

estimate the number of modes that can be supported in a planar dielectric waveguide that is 100m wide and has n1 = 1.490, n2 = 1.470 at the free-space source wavelength = 1m. Compare your estimation with the formula:

M = Int(2V/) + 1 Int(x): integer function

Vm 2

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Example 2.1.3: Mode field width (MFD), 2w0

The field distribution along y penetrates into the cladding as depicted in the following figure

The extent of the electric field across the guide is therefore more than 2a. Within the core, the field distribution is harmonic whereas from the boundary into the cladding, the field decays exponentially:

Ecladding (y) = Ecladding (0) exp (-claddingy)

Find the mode field width (MFD) 2w0 = 2a + 2, where = 1/ cladding

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2-2 Modal and Waveguide Dispersion in the Planar Waveguide

4/9/2009

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m

Slope = c/n2

Slope = c/n1

TE0

cut-off

TE1

TE2

Schematic dispersion diagram, vs. for the slab waveguide for various TEm. modes.cut ff corresponds to V = /2. The group velocity vg at any is the slope of the vs. curve at that frequency.


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What is important is the group velocity along the guide, the velocity at which the energy or information is transported.

The higher modes penetrate more into the cladding where the refractive index is smaller and the waves travel faster.

Waveguide condition

man mm =

cos)2(2 1

Waveguide Dispersion Diagram

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Waveguide Dispersion Diagram

The vs. m characteristicsThe slope at frequency is the group velocity vg

The group velocity at one frequency changes from one mode to another.For a given mode the group velocity changes with the frequency.

The cutoff frequency corresponds to the cutoff condition when

mdd

offcutc = 2=V

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m

Slope = c/n2

Slope = c/n1

TE0

cut-off

TE1

TE2

Schematic dispersion diagram, vs. for the slab waveguide for various TEm. modes.cut ff corresponds to V = /2. The group velocity vg at any is the slope of the vs. curve at that frequency.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)26

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Intermodal Dispersion

maxmin gg vL

vL

=

cnn

L21

Modal dispersion

Example: n1 = 1.48, n2 = 1.46

1/ 2 Half intensity points that is smaller than the full width

kmnsL

/67

Vg,min = c/n1Vg,max = c/n2

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

Intramodal Dispersion Fig. 10

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Intramodal Dispersion

Waveguide Dispersion

Material Dispersion: n = n() or n()

Dispersion in WG

- Intermodal dispersion: for multimode waveguide

- Intramodal dispersion: waveguide dispersionmaterial dispersion

Intermodal dispersion >> Intramodal dispersion

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2-3 Step Index Fiber

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n

y

n2 n1

Cladding

Core z

y

r

Fiber axis

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


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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


HE and EHHybrid modes

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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


)(exp),( ztjrEE lmlmLP =Linear polarized modes

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1

21

nnn

=

)(exp),( ztjrEE lmlmLP =

2/11

2/122

21 )2(

2)(2 == nnannaV

21

22

21

1

21

2nnn

nnn

=

405.2)(2 2/12221 == nn

aVc

offcut

2

2VM

22

21

22

2)/(nn

nkb

=

Normalized index difference

Linear polarized modes

Normalized frequency or V-number

Normalized index difference

Single mode cutoff frequency

Number of modes

Normalized propagation constant

typical

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Example 2.3.4: Group velocity and delay

Consider a single mode fiber with core and cladding indices of 1.448 and 1.440, core radius of 3m, operating at 1.5m. Given that we can approximate the fundamental mode normalized propagation constant b by

b (1.1428 0.996 / V )2 1.5 < V < 2.5

(1) Calculate the propagation constant .(2) Change the operating wavelength to by a small amount,

0.01%, and then recalculate the new propagation constant . (3) Then determine the group velocity vg of the fundamental mode at

1.5m, and the group delay g over 1 km of fiber.

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2-4 Numerical Aperture

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Cladding

Core < max

AB

< c

A

B

> c

> max

n0 n1

n2Lost

Propagates

Maximumacceptance anglemax is that whichjust givestotal internal reflectionat thecore-cladding interface, i.e.when=maxthen=c.Rays with>max (e.g. rayB) becomerefractedandpenetrate the claddingand areeventuallylost.

Fiber axis

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

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0

1max

)90sin(sin

nn

c

=

( )0

2/122

21

maxsin nnn

=

( ) 2/12221 nnNA =

0maxsin n

NA=

NAaV2

=

Numerical Aperture

Maximum acceptance angle max

Snells law -->

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Example 2.4.1: A multimode fiber and total acceptance angle

A step index fiber has a core diameter of 100m and a refractive index of 1.48. The cladding has a refractive index of 1.460. Calculate(a) the numerical aperture of the fiber, (b) acceptance angle from air, (c) number of modes sustainedwhen the source wavelength is 850nm.

Example 2.4.2: A single mode fiber

A typical single mode optical fiber has a core of diameter 8m and a refractive index of 1.46. The normalized index difference is 0.3%. The cladding diameter is 125m.(a) Calculate the numerical aperture of the fiber(b) Calculate the acceptance angle of the fiber (c) What is the single mode cut-off wavelength c of the fiber? 42

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2-5 Dispersion in Single Mode Fibers

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t

Spread,

t0

Spectrum,

1 2o

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, , 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.


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= mDL

2

2

dnd

cDm

dd

vgg

011 ==

Material Dispersion

Material Dispersion

Material Dispersion Coefficient

Fundamental Mode group delay time

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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, .


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Waveguide Dispersion Coefficient

= wDL

22

22

2)2(984.1

cnaN

D gw

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+= |DD|L wm

Chromatic Dispersion

Chromatic Dispersion or Total Dispersion

Chromatic Dispersion Coefficient wm

DDD +=

Dispersion Shifted Fiber (DSF): to design the waveguide dispersion to shift the zero (material) dispersion from 1330 nm to 1550 nm

by reducing the core radius and increasing the core doping

Dispersion flattened Fiber: to use multiple clad fiber to control the total chromatic dispersion that is flattened between two wavelength

DFF

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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 = Dm + 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.


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= pDL

Profile Dispersion: dispersion due to =()

Polarization Dispersion: dispersion due to an-isotropy, n1x-n1y

Polarization Modal Dispersion (PMD)

(originates from material dispersion)

Due to anisotropic composition, geometry, strain Different group delays even with monochromatic source

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Core

z

n1 x

// x

n1 y

// y

Ey

Ex

Ex

Ey

E

= Pulse spread

Input light pulse

Output light pulset

t

Intensity

Suppose that the core refractive index has different values along two orthogonaldirections corresponding to electric field oscillation direction (polarizations). We cantake x and y axes along these directions. An input light will travel along the fiber with Exand Ey polarizations having different group velocities and hence arrive at the output atdifferent times


Polarization modal dispersion

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Example 2.5.1: Material dispersion

By convention , the width of the wavelength spectrum of the source and the dispersion refer to half-power widths and not widths from one extreme end to the other.

1/2: width of intensity vs. wavelength spectrum between the half intensity points (i.e. linewidth)

1/2: width of the output light intensity vs. time signal between the half-intensity points

(a) Estimate the material dispersion effect per km of silica fiber operated from a light emitting diode (LED) emitting at 1.55m with a linewidth of 100nm.

(b) What is the material dispersion effect per km of silica fiber operated from a laser diode emitting at the same wavelength with a linewidth of 2nm?

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Example 2.5.2: Material, waveguide, and chromatic dispersion

Consider a single mode optical fiber with a core of SiO2- 13.5%GeO2 for which the material and waveguide dispersion coefficients are shown in the following figure. Suppose the fiber is excited from a 1.5m laser source with a linewidth 1/2 of 2 nm. (a) What is the dispersion per km of fiber if the core diameter 2a is 8 m?

(b) What should be the core diameter for zero chromatic dispersion at = 1.5m?

0

10

10

20

1.2 1.3 1.4 1.5 1.620

(m)

Dm

Dw

SiO2-13.5%GeO2

2.53.03.54.0a (m)

Dispersion coefficient (ps km-1 nm-1)

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2-6 Bit Rate, Dispersion, Electrical and Optical Bandwidth

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Bit rate capacity B (bits/sec) is related to the dispersion characteristics.The dispersion is measured by the pulse spread:

Full width at half power (FWHP)Full width at half maximum (FWHM)

There is no inter-symbol interference peak-to-peak separationthe maximum bit rate

Intuitive return-to-zero (RZ) bite rate (or data rate)

Non-return to zero (NRZ) bite rate 2B

21 /

2/1

5.0

B

Bit Rate and Dispersion

212 /

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t0

Emitter

Very shortlight pulses

Input Output

Fiber

PhotodetectorDigital signal

Information Information

t0

~2 1/2T

t

Output IntensityInput Intensity

1/2

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


RZ

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t

Output optical power

1/2

T = 41

0.50.61 2

A Gaussian output light pulse and some tolerable intersymbolinterference between two consecutive output light pulses (y-axis inrelative units). At time t = from the pulse center, the relativemagnitude is e-1/2 = 0.607 and full width root mean square (rms)spread is rms = 2.


607.02/1 =e 2rms =

NRZ

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21

590250

=..B

For a Gaussian pulse,

Root-mean-square deviation orRoot-mean-square (rms) dispersion

Full-width rms time spread: between the rms points of the pulse

Maximum RZ bit rate and rms dispersion

Maximum bit rate X distance

Total rms dispersion

2rms =

|D|.L.BL

ch

250250=

2intramodal

2intermodal

2 +=

( )( )

22

1 22

1

2

th t e

=

214250 = .

2

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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

19.075.0 Bfop

Optical and Electrical BandwidthOptical bandwidth for Gaussian dispersion

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19.075.0 Bfop

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Example 2.6.1: Bit rate and dispersion

Consider an optical fiber with a chromatic dispersion coefficient 8 ps km-1 nm-1 at an operation wavelength of 1.5m. Calculate (a) the bit rate distance product (BL),

(b) the optical and electrical bandwidths for a 10km fiber if a laser diode source with a FWHP linewidth 1/2of 2 nm is used.

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2-7 The Graded Index (GRIN) Optical Fiber

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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


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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'.


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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)


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arnn

ararnn

==

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.


4/23/2009

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0 2 4 6 8 10 12 14 16 18

Radius of curvature (mm)

103

102

101

1

10

102

B (m-1) for 10 cm of bend

= 633 nm = 790 nmV 2.08V 1.67

Measured microbending loss for a 10 cm fiber bent by different amounts of radius ofcurvature R. Single mode fiber with a core diameter of 3.9 m, cladding radius 48 m, = 0.004, NA = 0.11, V 1.67 and 2.08 (Data extracted and replotted with correctionfrom, A.J. Harris and P.F. Castle, IEEE J. Light Wave Technology, Vol. LT14, pp. 34-40, 1986; see original article for discussion of peaks in B vs. R at 790 nm).


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Example 2.9.1: Rayleigh scattering limit

What is the attenuation due to Rayleigh scattering at around the = 1.55m window given that pure silica (SiO2) has the following properties: Tf = 1730C (softening temperature);

T = 7 x 10-11 m2N-1 (at high temperatures); n = 1.4446 at 1.5m.

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Example 2.9.1: Rayleigh scattering limit

What is the attenuation due to Rayleigh scattering at around the = 1.55m window given that pure silica (SiO2) has the following properties: Tf = 1730C (softening temperature);

T = 7 x 10-11 m2N-1 (at high temperatures); n = 1.4446 at 1.5m.

( )

( ) ( ) ( )( )( )

( )( ) 112

1215

23112246

3

224

3

142.01027.334.434.4

1027.31027.3

273730.11038.110714446.11055.13

8

138

===

==

+

dBkmkmnAttenuatio

kmm

Tkn

RdB

fBTR

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Example 2.9.2: Attenuation along an optical fiber

The optical power launched into a single-mode fiber from a laser diode is approximately 1 mW. The photodetector at the output required a minimum power of 10 nW to provide a clear signal (above noise). The fiber operates at 1.3m and has an attenuation coefficientof 0.4 dB km-1. What is the maximum length of fiber that can be used without inserting a repeater (to regenerate the signal)?

kmPPL

PP

L

out

in

dB

out

indB

1251010log10

4.01log101

log101

8

3

=

=

=

=

The signal has to be amplified after a distance of about 50 ~ 100km, and eventually regenerated by using a repeater.

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2-10 Fiber Manufacture

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Preform feed

Furnace 2000

Thicknessmonitoring gauge

Take-up drum

Polymer coater

Ultraviolet light or furnacefor curing

Capstan

Schematic illustration of a fiber drawing tower.


Fiber Drawing

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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)

n1n2


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Vapors: SiCl 4 + GeCl 4 + O 2

Rotate mandrel

(a)

Deposited sootBurner

Fuel: H 2

Target rod

Deposited Ge doped SiO 2

(b)

Furnace

Porous sootpreform with hole

Clear solidglass preform

Drying gases

(c)

Furnace

Drawn fiber

Preform

Schematic illustration of OVD and the preform preparation for fiber drawing. (a)Reaction of gases in the burner flame produces glass soot that deposits on to the outsidesurface of the mandrel. (b) The mandrel is removed and the hollow porous soot preformis consolidated; the soot particles are sintered, fused, together to form a clear glass rod.(c) The consolidated glass rod is used as a preform in fiber drawing.


Outside Vapor Deposition (OVD)

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Example 2.10.1: Fiber drawing

In a certain fiber production process a preform of length 110cm and diameter 20 mm is used to draw a fiber. Suppose that the fiber drawing rate is 5 m/s. What is the maximum length of the fiber that can be drawn from this preform if the last 10 cm of the preform is not drawn and the fiber diameter is 125 m? How long does it take to draw the fiber?

( )( )( )

( )( ) .4.1/6060/525600

)/()()(

/5

6.252560010125

10201.01.126

23

22

hrshrssm

mhrkmRate

kmLengthhrsTime

smrate

kmmm

mmL

dLdL

f

ppff

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Typical drawing rates are in the range 5 20m/s so that 1.4hrs would be on the long-side.

chap2 dielectric waveguides and optical...

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