Optic All Lll Lllll

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3.1 THE STRUCTURE OF OPTICAL FIBERS3.1 THE STRUCTURE OF OPTICAL FIBERS

3.2 MODES OF AN OPTICAL FIBER3.2 MODES OF AN OPTICAL FIBER

3.3 NUMERICAL APERTURE AND3.3 NUMERICAL APERTURE ANDACCEPTANCE ANGLEACCEPTANCE ANGLE

3.4 DISPERSION IN OPTICAL FIBERS3.4 DISPERSION IN OPTICAL FIBERS

3.5 SINGLE3.5 SINGLE --MODE FIBERS: MODE PROFILE,MODE FIBERS: MODE PROFILE,MODEMODE --FIELD DIAMETER, AND SPOT SIZEFIELD DIAMETER, AND SPOT SIZE

3.6 NORMALIZED FREQUENCY,3.6 NORMALIZED FREQUENCY,NORMALIZED PROPAGATION CONSTANT,NORMALIZED PROPAGATION CONSTANT,AND CUTOFF WAVELENGTHAND CUTOFF WAVELENGTH

StepStep --Index Optical Fiber Index Optical Fiber – – Typically the core of a singleTypically the core of a single --mode fiber will bemode fiber will be

of the order of 2of the order of 2 --1010 μ mm ..

Figure 3.1 The structure of the step-index optical fiber



Graded Graded - - Ind ex Optical Ind ex Optical Fiber Fiber

– – Communications fibersCommunications fibersfall into this category withfall into this category withΔΔ usually being lessusually being lessthan 3 %than 3 %

– – For many applications,For many applications, αα= 2 is the optimum (i.e. a= 2 is the optimum (i.e. aparabolic profile)parabolic profile) Figure 3.2 Refractive index profile

of the graded-index optical fiber core

α

Δ−=ar nr n 211 ar ≤

( ) Δ−= 211nr n ar ≥1

21

nnn −

=Δ

TETE lmlm or or TMTM lmlm modesmodes – – The integer The integer l l represents the fact that there willrepresents the fact that there will

bebe 22l l field maxima around the circumferencefield maxima around the circumferenceof the field distribution.of the field distribution.

– – The integer The integer mm refers to therefers to the mm field maximafield maximaalong a radius.along a radius.

Meridional Meridional raysraysThe propagation angles of the skew rays areThe propagation angles of the skew rays are

such that it is possible for components of bothsuch that it is possible for components of boththethe E E andand H H fields to be transverse to the fiber fields to be transverse to the fiber axis.axis.

HEHE lmlm or or EHEH lmlm depending on whether thedepending on whether the E E or or H H field dominates the transverse field.field dominates the transverse field.InIn weakly guid ing f ib ers weakly guid ing f ibers , the exact modal, the exact modal

solutions are usually approximated bysolutions are usually approximated by Linearly Linearly Polar ized mo des Polar ized mo des , designated, designated LPLP lmlm modesmodes



Figure 3.3 Propagation constant of the exact fiber modes plotted against normalized frequency.

Figure 3.4 Intensityprofiles of the threelowest-order LPmodes.

Reproduced from G.P. Agrawal (1997) Fiber Optic Communica-tions Systems, 2nd edn, John Wiley & Sons, New York by permission of John Wiley and Sons Inc.

HE l +1, m, EH l -1,mLP lm(l ≠ 0 or 1)

HE 2m, TE 0m, TM 0mLP lm

HE 22, TE 02, TM 02LP 12

HE 41, EH 21LP 31

HE 12LP 02

HE 31, EH 11LP 21

HE 21, TE 01, TM 01LP 11

HE 11LP 01

Exact modesLinearly polarized

Table 3.1 Relationship between approximate LP modes and exactmodes. Source: Optical Fiber Communications, Principles and Practice, 2nd edn, J M Senior, Pearson Education Limited.Reproduced by permission of Pearson Education

Consider a graded Cons id er a graded - - index f iber wi th a index f iber wi th a parabo lic refract ive ind ex pro fi le (i .e .parabo lic refract ive ind ex pro fi le (i .e . α = = 22 ) )

Figure 3.5 Structure of the graded-index optical fiber

α

Δ−=ar

nr n 211 ar ≤

( ) Δ−= 211nr n ar ≥

Figure 3.6 Total internal reflection in a graded-index fiber. Source:Optical Fiber Communications, Principles and Practice, 2nd edn, J MSenior, Pearson Education Limited. Reproduced by permission of Pearson Education



Figure 3.7 Mode trajectories in graded-index fiber. Source: Optical Fiber Commu-nications, Principles and Practice, 2nd edn, J M Senior,Pearson Education Limited. Reproduced by permission of PearsonEducation

NUMERICAL APERTURE AND ACCEPTANCE ANGLENUMERICAL APERTURE AND ACCEPTANCE ANGLE

Critical angleCritical angle θθ cc, Acceptance angle, Acceptance angle θθ aa ..

Figure 3.8 Acceptance angle of an optical fiber

cccaa nnnn θ θ θ θ 2111 sin1cos90sinsin −==−=

1

2sinnn

c=θ


Numerical ApertureNumerical Aperture

Figure 3.8 Acceptance angle of an optical fiber

22

21

21

221 /1sin nnnnnn aa

−=−=θ

22

21sin nnn NA aa

−== θ Δ= 21n NA




For skew raysFor skew rays

Figure 3.9 The acceptance angle of skew rays. Source: Optical Fiber Communications, Principles and Practice, 2nd edn, J M Senior, PearsonEducation Limited. Reproduced by permission of Pearson Education

22

21cossin nnn NA aa

−== γ θ Dispersion in optical fibers means that parts of Dispersion in optical fibers means that parts of the signal propagate through the fiber at slightlythe signal propagate through the fiber at slightlydifferent velocities, resulting in distortion of thedifferent velocities, resulting in distortion of thesignal.signal.

t 0

Emitter

Very shortlight pulses

Input Output

Fiber

Photodetector Digital signal

Information Information

t 0

~2 τ 1/ 2

T

t

Output IntensityInput Intensity² τ 1/ 2

~2δτ

δτ

For a given pulse broadening,For a given pulse broadening, .. If pulses are not toIf pulses are not tooverlap, then the maximum pulse broadening must be aoverlap, then the maximum pulse broadening must be amaximum of half of the transmission period,maximum of half of the transmission period, T T ..

The bit rateThe bit rate B BT T ,, isis 1/1/T T

Some texts use a more conservative rule of thumb,Some texts use a more conservative rule of thumb,makingmaking B BT T ≤≤ ¼¼For aFor a gaussiangaussian pulse, with anpulse, with an r.m.sr.m.s . width. width σσ .. A typical A typicalrule of thumb isrule of thumb is B BT T ≤≤ 0.2/0.2/ σσ

T 21

≤δτ

δτ

δτ 21≤

T B

δτ



IntermodalIntermodal dispersion is a result of the differentdispersion is a result of the differentpropagation times of different modes (differentpropagation times of different modes (different mm) within a) within afiber.fiber.

Low order modeHigh order mode

Cladding

Core

Light pulse

t 0 t

Spread, Δ τ

Broadenedlight pulse

IntensityIntensity

Axial

Schematic illustration of light propagation in a slab dielectric waveguide. Light pulse

entering 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.

The fastest and slowest modes possible in such a fiber The fastest and slowest modes possible in such a fiber will be the axial ray, and the ray propagating at the criticalwill be the axial ray, and the ray propagating at the criticalangleangle θθ c c . ( . ( sinsin θθ c c == nn22//nn11 ) )

Figure 3.10 The origin of modal dispersion

In a multimode fiber, generallyIn a multimode fiber, generally

c

Ln

nc

Lt 1

1min

/velocity

distance ===

c

c nc L

nc L

t θ

θ sin/

sin/ 1

1max

==

2

2

1

1max /

sin/nn

c L

nc Lt c == θ

or



ΔΔ is the relative refractive indexis the relative refractive index ((nn11--nn22)/)/nn11

For nFor n 11 ≈≈ nn 22

If we letIf we let ΔΔ = 0.01 and= 0.01 and nn11 = 1.49. Hence= 1.49. Hence δτ δτ ==49.6 ns/km. The maximum bit rate becomes49.6 ns/km. The maximum bit rate becomes

≈≈ 1010 Mbit/sMbit/s ··kmkm

Δ=

−

=−=−=2

21

1

21

2

211

2

21

minmax nn

c L

nnn

nn

c L

c Ln

nn

c L

t t t siδ

c Ln

t si

Δ= 1δ

δτ 21=

T B

n1

n2

21

3

nO

n1

21

3

n

n2

OO' O''

n2

(a) Multimode stepindex fiber. Ray paths

are different so thatrays arrive at differenttimes.

(b) Graded index fiber.Ray paths are different

but so are the velocitiesalong the paths so thatall the rays arrive at thesame time.

23

For gradedFor graded --index multimode fiber, the velocity of the rayindex multimode fiber, the velocity of the rayis inversely proportional to the local refractive index, itis inversely proportional to the local refractive index, itcan be shown thatcan be shown that

If we letIf we let ΔΔ = 0.01 and= 0.01 and nn11 = 1.49. Hence= 1.49. Hence δτ δτ = 62= 62 psps /km./km.The maximum bit rate becomesThe maximum bit rate becomes

≈≈ 88 Gbit/sGbit/s ··kmkm

c Ln

t gi 8

21Δ

=δ

δτ 21=

T B



Different spectral componentsDifferent spectral componentsof the light source may haveof the light source may havedifferent propagation delays,different propagation delays,and hence pulse broadeningand hence pulse broadeningmay occur, even if transmittedmay occur, even if transmittedby a single mode.by a single mode.

y

E ( y)

Cladding

Cladding

Core

λ 2

> λ 1λ 1 > λ c

ω 2 < ω 1ω 1 < ω cut-off

v g 1

y

v g 2 > v g 1

τ t

Spread, τ

t 0

λ

Spectrum, λ

λ 1 λ 2λ o

Intensity Intensity Intensity

Cladding

CoreEmitter

Very shortlight pulse

v g (λ 2 )

v g (λ 1 )Input

Output

Δλδτ

Phase velocityPhase velocity

Group velocityGroup velocity

Traveling time required for each frequencyTraveling time required for each frequency(wavelength) component(wavelength) component

HenceHence

ThusThus

β ω /= pv

δβ δω /= g v

g v L /=τ

2

2

ω β

ω τ

∂∂=

∂∂ L

∂

∂=

∂

∂

g v L

1ω ω

τ

λ 1

λ 2

For λ 1 <λ 2

λ1

λ 2

Vg

Vp 2

Vp 1

λ 1 & λ 2

λ 1 & λ 2

Zero dispersion D = 0, Vp 1 = Vp 2 (Light Packet at the same speed Vg = Vp )

Normal dispersion D < 0, Vp 1 < Vp 2 (Slow Light Packet Vg < Vp )

Anomalous dispersion D > 0, Vp 1 > Vp 2 (Fast Light Packet Vg > Vp )

For λ 1 <λ 2



IntramodalIntramodal (Chromatic) Dispersion(Chromatic) Dispersionin Singlein Single --mode Optical Fiber mode Optical Fiber

For dispersion flattened singleFor dispersion flattened single --mode optical fiber,mode optical fiber,operated near operated near λλ ZDZD , e.g., e.g. D D == 11 ps/(nmps/(nm ··kmkm ))

≈≈ 500500 Gbit/sGbit/s ··kmkm

ComparisonComparison

SI MM fiber SI MM fiber B BT T ≈≈

1010 Mbit/sMbit/s ··kmkmGI MM fiber GI MM fiber B BT T ≈≈ 88 Gbit/sGbit/s ··kmkmSI SM fiber SI SM fiber B BT T ≈≈ 500500 Gbit/sGbit/s ··kmkm

chδτ =

T B

SINGLESINGLE --MODE FIBERS: MODE PROFILE,MODE FIBERS: MODE PROFILE,MODEMODE --FIELD DIAMETER, AND SPOT SIZEFIELD DIAMETER, AND SPOT SIZE

Cylindrical coordinatesCylindrical coordinates((r,r, φ φ andand z z )) – – expexp j jββ z z

– – expexp jm jmφ φ – – J J mm ((r r ),), K K mm ((r r ))

Cartesian coordinatesCartesian coordinates(( x, y x, y andand z z )) – – expexp j jββ z z

– – expexp jmx jmx or or indepindep . of . of x x – – coscos k k ym ym y y, exp, exp --k k ym ym y y

Figure 3.13 (a) Variation of the first four Bessel functions. (b) Variation of thefirst two modified Bessel functions.

)(exp),( z t jr E E lmlm LP β ω ϕ −= )(exp)( z t j y E E mmm β ω −=



MODE PROFILEMODE PROFILE – – J J 00((r r ),), K K 00 ((r r ))

Figure 3.14 Field shape of the fundamental mode of a step-index fiber, for normalized frequencies of V = 1.5 and V = 2.4.

Mode Field Diameter =Mode Field Diameter = 22 w w 00

Spot size =Spot size = w w 00

Figure 3.15 Approximation to the fundamental mode, showing the mode-fielddiameter (MFD) and the spot size w 0

V V

aw1

22 0

+=

Normalized Frequency:Normalized Frequency: V V number n u m b e r

Norm alized propagat ion cons tant : Norm alized propagat ion constant : b b

aa : f iber core radius : f iber core radius

Δ : re la t ive refract ive ind ex difference : re la t ive refract ive ind ex difference

λλ 00 operat ing wavelength operat ing wavelength

Δ=−= 222

10

22

21

0

annnaV λ π

λ π

Δ

−=

−

−=

21

22

20

22

21

22

20

2//

nnk

nnnk

bβ β



Figure 3.16 Propagation constant of the exact fiber modesplotted against normalized frequency.

2.405

Fitting curve for LPFitting curve for LP 0101

CutCut --off Condition for Single Mode Fiber off Condition for Single Mode Fiber V V cc = 2.405 for step= 2.405 for step --index fibersindex fibers

λλcc//λλ

00== V V //V V

cc..

Δ= 22

1c

c anV π λ

405.20

cλ λ

V =

)5.25.1(996.0

1428.12

<<

−≈ V V

b

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