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Fiber-Optic Communication Systems

Govind P. AgrawalInstitute of OpticsUniversity of RochesterRochester, NY 14627email: gpa@optics.rochester.edu

c©2003 G. P. Agrawal

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Course Outline

• Introduction

• Fiber Loss, Dispersion, and Nonlinearities

• Receiver Noise and Bit Error Rate

• System Design and Performance

• Loss Management: Optical Amplifiers

• Dispersion Management

•Multichannel Lightwave Systems

• Optical Solitons

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Historical Perspective

• Smoke signals; <1500

• Semaphore Devices; 1500-1800

•Mechanical Coding (Chappe); 1792

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Historical PerspectiveElectrical Era

• Telegraph; 1836

• Telephone; 1876

• Coaxial Cables; 1840

• Microwaves; 1948

Optical Era

• Optical Fibers; 1978

• Optical Amplifiers; 1990

• WDM Technology; 1996

• Multiple bands; 2002

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Communication Technologies

Performance criterion: B×L

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Information Revolution• Industrial revolution of 19th century gave way to

information revolution during the 1990s.

• Internet made the rapid distribution of information possible.

• Fiber-Optic Revolution is a natural consequence of the

Internet growth .

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Five Generations1. 0.8-µm systems (1980); Graded-index fibers

2. 1.3-µm systems (1985); Single-mode fibers

3. 1.55-µm systems (1990); Single-mode lasers

4. WDM systems (1996); Optical amplifiers

5. L and S bands (2001); Raman amplification

1980 1985 1990 1995 2000 2005Year

0.01

0.1

1

10

100

1000

10000

Bit

Rat

e (G

b/s) Research

Commercial

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Basic ConceptsAnalog and Digital Signals

• Lightwave Systems use digital format.

• Optical signal is a stream of 0 and 1 bits.

• Bit rate B determines the time slot TB = 1/B for each bit.

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Analog to Digital Conversion

• Sampling: fs≥ 2∆ f (sampling theorem).

• Quantization: M > Amax/AN (error < noise).

• Coding: M = 2m; m bits/sample (Binary coding).

• Bit rate : B = m fs≥ (2∆ f ) log2M > (∆ f/3)SNR.

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Audio and Video SignalsDigital Audio Signal

• ∆ f = 3.1 kHz (0.3 to 3.4 kHz); SNR = 30 dB.

• Minimum B = (∆ f/3)SNR = 31 kb/s.

• In practice, B = 64 kb/s ( fs = 8 kHz; 8 bits/sample).

Digital Video Signal

• ∆ f = 4 MHz; SNR = 50 dB.

• Minimum B = (∆ f/3)SNR = 66 Mb/s.

• In practice, B = 100Mb/s ( fs = 10 MHz; 10 bits/sample).

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Channel MultiplexingTDM : Time-division multiplexing

FDM : Frequency-division multiplexing

Optical FDM = WDM (wavelength-division multiplexing)

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Evolution of Standards• No standards until 1988.

• US standard: synchronous optical network(SONET).

• ITU standard: synchronous digital hierarchy(SDH).

SONET SDH B (Mb/s) Channels

OC-1 51.84 672

OC-3 STM-1 155.52 2,016

OC-12 STM-4 622.08 8,064

OC-48 STM-16 2,488.32 32,256

OC-192 STM-64 9,953.28 129,024

OC-768 STM-256 39,813.12 516,096

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Modulation FormatsOptical Carrier

E(t) = eAcos(ω0t +φ)

• Amplitude-shift keying (ASK): modulate A

• Frequency-shift keying (FSK): modulate ω0

• Phase-shift keying (PSK): modulate φ

• Polarization-shift keying (PoSK): modulate e

Most lightwave systems employ ASK;

also called on–of keying (OOK) or

Intensity modulation with Direct Detection (IM/DD)

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Optical Bit Stream• Return-to-zero (RZ)

• nonreturn-to-zero (NRZ)

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Lightwave System ComponentsGeneric System

Transmitter Design

Receiver Design

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Optical Fibers• Most suitable as communication channel because of dielectric waveg-

uiding (acts like an optical wire).

• Total internal reflection at the core-cladding interface.

• Single-mode propagation for core size < 10 µm.

What happens to Signal?

• Fiber losseslimit the transmission distance (minimum loss near

1.55 µm).

• Chromatic dispersion limits the bit rate through pulse broadening.

• Nonlinear effects distort the signal and limit the system perfor-

mance.

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Fiber Fundamentals

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Total internal reflection• Refraction at the air–glass interface: n0sinθi = n1sinθr

• Total internal reflection at the core-cladding interface

if φ > φc = sin−1(n2/n1).

Numerical Aperture: Maximum angle of incidence

n0sinθmaxi = n1sin(π/2−φc) = n1cosφc =

√n2

1−n22

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Modal Dispersion• Multimode fibers suffer from modal dispersion.

• Shortest path length Lmin = L (along the fiber axis).

• Longest path length for the ray close to the critical angle

Lmax = L/sinφc = L(n1/n2).

• Pulse broadening: ∆T = (Lmax−Lmin)(n1/c).

• Modal dispersion: ∆T/L = n21∆/(n2c).

• Limitation on the bit rate

∆T < TB = 1/B; B∆T < 1; BL <n2c

n21∆

.

• Single-mode fibersessential for high performance.

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Graded-Index Fibers

• Refractive index n(ρ) =

n1[1−∆(ρ/a)α ]; ρ < a,

n1(1−∆) = n2 ; ρ ≥ a.

• Ray path obtained by solving d2ρ

dz2= 1

ndndρ

.

• For α = 2, ρ = ρ0cos(pz)+(ρ ′0/p)sin(pz).

• All rays arrive simultaneouslyat periodic intervals.

• Limitation on the Bit Rate: BL < 8cn1∆2.

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Fiber Modes• Shape-preserving solutions of Maxwell’s equations

∇×E =−∂B/∂ t

∇×H = ∂D/∂ t

∇ ·D = 0

∇ ·B = 0

• Constitutive Relations

D = ε0E+P

B = µ0H +M

• Linear Susceptibility

P(r , t) = ε0

∫ ∞

−∞χ(r , t− t ′)E(r , t ′)dt′

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Fiber Modes (cont.)• It is easier to work in the Fourier domain:

E(r ,ω) =∫ ∞

−∞E(r , t)exp(iωt)dt

• Helmholtz Equation: Each frequency component satisfies

∇2E+n2(ω)k20E = 0.

k0 = ω/c = 2π/λ , n(ω) = (1+Reχ)1/2.

• n = n1 inside the core but changes to n2 in the cladding.

• Frequency dependence of n responsible for dispersion.

• Material absorption governed by Im χ .

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Fiber Modes (cont.)• Any solution must satisfy boundary conditions at the

core–cladding interface (x2+y2 = a2).

• Useful to work with cylindrical coordinates ρ,φ ,z.

• Only two components of E and H are independent.

• Common to choose Ez and Hz as independent components.

• Equation for Ez in cylindrical coordinates:

∂ 2Ez

∂ρ2+

1ρ

∂Ez

∂ρ+

1ρ2

∂ 2Ez

∂φ 2+

∂ 2Ez

∂z2+n2k2

0Ez = 0.

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Fiber Modes (cont.)• Use the method of separation of variables:

Ez(ρ,φ ,z) = F(ρ)Φ(φ)Z(z).

• We then obtain three ODEs:

d2Z/dz2+β2Z = 0,

d2Φ/dφ2+m2Φ = 0,

d2Fdρ2

+1ρ

dFdρ

+(

n2k20−β

2−m2

ρ2

)F = 0.

• β and m are two constants (m must be an integer).

• First two equations can be solved easily to obtain

Z = exp(iβz), Φ = exp(imφ).

• F(ρ) satisfies the Bessel equation.

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Fiber Modes (cont.)• General solution for Ez and Hz:

Ez =

AJm(pρ)exp(imφ)exp(iβz) ; ρ ≤ a,

CKm(qρ)exp(imφ)exp(iβz); ρ > a.

Hz =

BJm(pρ)exp(imφ)exp(iβz) ; ρ ≤ a,

DKm(qρ)exp(imφ)exp(iβz); ρ > a.

where p2 = n21k

20−β 2, q2 = β 2−n2

2k20.

• Boundary conditions: Ez, Hz, Eφ , and Hφ should be continuous

across the core–cladding inteface (ρ = a).

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Fiber Modes (cont.)Eigenvalue Equation[

J′m(pa)pJm(pa)

+K′

m(qa)qKm(qa)

][J′m(pa)pJm(pa)

+n2

2

n21

K′m(qa)

qKm(qa)

]

=m2

a2

(1p2

+1q2

)(1p2

+n2

2

n21

1q2

)where p2 = n2

1k20−β 2 and q2 = β 2−n2

2k20.

• This equation should be solved numerically.

• Multiple solutions for β for a given fiber.

• Each solution represents an optical mode.

• Number of modes governed by V = k0a√

n21−n2

2.

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Fiber Modes (cont.)

• Effective mode indexn = β/k0 between n1 and n2.

• Useful to introduce a normalized quantity

b = (n−n2)/(n1−n2), (0 < b < 1).

• Modes quantified through β (ω) or b(V).

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Single-Mode Condition• Modes denoted as HEmn or EHmn, where n = 1,2, . . . in decreasing

order of the eigenvalues βmn.

• Also labelled as TE0n and TM0n for m= 0.

• Number of modes increases rapidly with V paprameter.

• A mode ceases to exist when q = 0 (no decay in the cladding).

• TE01 and TM01 reach cutoff when J0(V) = 0.

• Single-mode fibersrequire V < 2.405.

• Typically core radius a < 5 µm and n1−n2 < 0.005.

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Single-Mode Fibers• Fibers support only the HE11 mode when the core size is such that

V = k0a(n21−n2

2) < 2.405.

• This mode is almost linearly polarized (|Ez|2 |ET|2).

• Spatial mode distribution approximately Gaussian

Ex(x,y,z,ω) = A0(ω)exp

(−x2+y2

w2

)exp[iβ (ω)z].

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Single-Mode Properties• Spot size: w/a≈ 0.65+1.619V−3/2+2.879V−6.

• Mode index:

n = n2+b(n1−n2)≈ n2(1+b∆),

b(V)≈ (1.1428−0.9960/V)2.

• Confinement factor:

Γ =Pcore

Ptotal=∫ a

0 |Ex|2ρ dρ∫ ∞0 |Ex|2ρ dρ

= 1−exp

(−2a2

w2

).

• Γ≈ 0.8 for V = 2 but drops to 0.2 for V = 1.

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Fiber Birefringence• Real fibers exhibit some birefringence (nx 6= ny).

• Modal birefringence quite small (Bm = |nx− ny| ∼ 10−6).

• Beat length: LB = λ/Bm.

• State of polarization evolves periodically.

• Birefringence varies randomly along fiber length (PMD) because

of stress and core-size variations.

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Fiber DispersionOrigin: Frequency dependence of the mode index n(ω):

β (ω) = n(ω)ω/c = β0+β1(ω−ω0)+β2(ω−ω0)2+ · · · ,

where ω0 is the carrier frequency of the optical pulse.

• Transit time for a fiber of length L : T = L/vg = β1L.

• Different frequency components travel at different speeds and arrive

at different times at output end (pulse broadening).

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Fiber Dispersion (continued)Pulse broadening governed by group-velocity dispersion:

∆T =dTdω

∆ω =d

dω

Lvg

∆ω = Ldβ1

dω∆ω = Lβ2∆ω,

where ∆ω is pulse bandwidth and L is fiber length.

GVD parameter: β2 =(

d2β

dω2

)ω=ω0

.

Alternate definition : D = ddλ

(1vg

)=−2πc

λ 2 β2.

Limitation on the bit rate : ∆T < TB = 1/B, or

B(∆T) = BLβ2∆ω ≡ BLD∆λ < 1.

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Material Dispersion• Refractive index of of any material is frequency dependent (chro-

matic dispersion).

• Material dispersion for silica is governed by the Sellmeier equation

n2(ω) = 1+M

∑j=1

B jω2j

ω2j −ω2

.

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Waveguide Dispersion• Mode index n(ω) = n2(ω)+δnW(ω).

• Material dispersion DM results from n2(ω) (silica index).

• Waveguide dispersion DW results from δnW(ω) and depends on

core sizeand dopant distribution .

• Total dispersion D = DM +DW can be controlled.

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Higher-Order Dispersion• dispersive effects do not disappear at λ = λZD.

• D cannot be made zero at all frequencies within the pulse spectrum.

• Higher-order dispersive effects are governed by the dispersion slopeS= dD/dλ .

• S can be related to third-oder dispersion β3 as

S= (2πc/λ2)2

β3+(4πc/λ3)β2.

• At λ = λZD, β2 = 0, and S is proportional to β3.

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Polarization-Mode Dispersion• Real fibers exhibit some birefringence (nx 6= ny).

• Orthogonally polarized components of a pulse travel at different

speeds. The relative delay is given by

∆T =∣∣∣∣ Lvgx− L

vgy

∣∣∣∣= L|β1x−β1y|= L(∆β1).

• Birefringence varies randomly along fiber length (PMD) because

of stress and core-size variations.

• RMS Pulse broadening :

σT ≈ (∆β1)√

2lcL≡ Dp

√L.

• PMD parameter Dp∼ 0.01–10 ps/√

km

• PMD can degrade the system performance considerably (especially

for old fibers).

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Commercial Fibers

Fiber Type and Aeff λZD D (C band) SlopeSTrade Name (µm2) (nm) ps/(km-nm) ps/(km-nm2)Corning SMF-28 80 1302–1322 16 to 19 0.090Lucent AllWave 80 1300–1322 17 to 20 0.088Alcatel ColorLock 80 1300–1320 16 to 19 0.090Corning Vascade 101 1300–1310 18 to 20 0.060TrueWave-RS 50 1470–1490 2.6 to 6 0.050Corning LEAF 72 1490–1500 2 to 6 0.060TrueWave-XL 72 1570–1580 −1.4 to−4.6 0.112Alcatel TeraLight 65 1440–1450 5.5 to 10 0.058

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Pulse Propagation Equation• Optical Field at frequency ω at z= 0:

E(r ,ω) = xF(x,y)B(0,ω)exp(iβz).

• Optical field at a distance z:

B(z,ω) = B(0,ω)exp(iβz).

• Expand β (ω) is a Taylor series around ω0:

β (ω) = n(ω)ω

c≈ β0+β1(∆ω)+

β2

2(∆ω)2+

β3

6(∆ω)3.

• Introduce Pulse envelope:

B(z, t) = A(z, t)exp[i(β0z−ω0t)].

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Pulse Propagation Equation• Pulse envelope is obtained using

A(z, t) =1

2π

∫ ∞

−∞d(∆ω)A(0,∆ω)exp

[iβ1z∆ω +

i2

β2z(∆ω)2 +i6

β3z(∆ω)3− i(∆ω)t].

• Calculate ∂A/∂z and convert to time domain by replacing

∆ω with i(∂A/∂ t).

• Final equation:

∂A∂z

+β1∂A∂ t

+iβ2

2∂ 2A∂ t2

− β3

6∂ 3A∂ t3

= 0.

• With the transformation t ′ = t−β1z and z′ = z, it reduces to

∂A∂z′

+iβ2

2∂ 2A∂ t ′2

− β3

6∂ 3A∂ t ′3

= 0.

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Pulse Propagation Equation• If we neglect third-order dispersion, pulse evolution is governed by

∂A∂z

+iβ2

2∂ 2A∂ t2

= 0.

• Compare with the paraxial equation governing diffraction:

2ik∂A∂z

+∂ 2A∂x2

= 0.

• Slit-diffraction problem identical to pulse propagation problem.

• The only difference is that β2 can be positive or negative.

• Many results from diffraction theory can be used for pulses.

• A Gaussian pulse should spread but remain Gaussian in shape.

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Pulse BroadeningChirped Gaussian Pulse

A(0, t) = A0exp

[−(1+ iC)t2

2T20

].

• Input pulse width TFWHM = 2(ln2)1/2T0≈ 1.665T0.

• Input chirp : δω(t) =−∂φ

∂ t = CT2

0t.

• Pulse spectrum

A(0,ω) = A0

(2πT2

0

1+ iC

)1/2

exp

[− ω2T2

0

2(1+ iC)

]• Spectral width: ∆ω0 =

√1+C2/T0.

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Pulse Broadening (cont.)Optical Pulse after propagation (β3 = 0):

A(z, t) =A0√Q(z)

exp

[−(1+ iC)t2

2T20 Q(z)

], Q(z) = 1+(C− i)

β2z

T20

.

• Pulse maintains its Gaussian shape.

• Its chirp changes to C1(z) = C+(1+C2)β2z/T20 .

• Its width is given by

T1

T0=

[(1+

Cβ2z

T20

)2

+(

β2z

T20

)2]1/2

.

• Unchirped pulse broadens by a factor√

1+(z/LD)2.

• LD = T20 /|β2| is called the dispersion length.

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Effect of Chirp

• Broadening depends on the sign of β2C.

• Pulse compressesinitially if β2C < 0.

• Pulse becomes shortest and unchirped at a certain distance:

zmin =[|C|/(1+C2)

]LD, Tmin

1 = T0/(1+C2)1/2.

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Third-Order Dispersion• When β2 = 0, pulse broadening depends on β3.

• Analytic solution exists in terms of an Airy function.

• Pulse does not maintain its Gaussian shapeand develops consid-

erable structure (subpulses near an edge).

• A proper measure of the pulse width is the RMS width defined as

σ =[< t2 >−< t >2

]1/2,

< tm >=∫ ∞−∞ tm|A(z, t)|2dt∫ ∞−∞ |A(z, t)|2dt

.

• Broadening factor:

σ2

σ20

=(

1+Cβ2L

2σ20

)2

+(

β2L

2σ20

)2

+(1+C2)2

(β3L

4√

2σ30

)2

.

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Source Spectral Width• Source spectral width can exceed pulse spectral width when a low-

coherence source (LED) is used.

• Introduce a parameter Vω = 2(∆ωs)σ0 for a source with Gaussian

spectrum of RMS width ∆ωs.

• RMS pulse width then increases as

σ2

σ20

=(

1+Cβ2L

2σ20

)2

+(1+V2ω)(

β2L

2σ20

)2

+(1+C2+V2ω)2

(β3L

4√

2σ30

)2

.

• Quite useful for analyzing the GVD effects.

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Dispersion LimitationsLarge Source Spectral Width: Vω 1

• Assume input pulse to be unchirped (C = 0).

• Set β3 = 0 when λ 6= λZD:

σ2 = σ

20 +(β2Lσω)2≡ σ

20 +(DLσλ)2.

• 96% of pulse energy remains within the bit slot if 4σ < TB = 1/B.

• Using 4Bσ ≤ 1, and σ σ0, BL|D|σλ ≤ 14.

• Set β2 = 0 when λ = λZD to obtain

σ2 = σ

20 + 1

2(β3Lσ2ω)2≡ σ2

0 + 12(SLσ2

λ)2.

• Dispersion limit: BL|S|σ2λ≤ 1/

√8.

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Dispersion LimitationsSmall Source Spectral Width: Vω 1

• When β3 = 0 and C = 0,

σ2 = σ

20 +(β2L/2σ0)2.

• One can minimize σ by adjusting input width σ0.

• Minimum occurs for σ0 = (|β2|L/2)1/2 and leads to σ = (|β2|L)1/2.

• Dispersion limit when β3 = 0:

B√|β2|L≤ 1

4.

• Dispersion limit when β2 = 0:

B(|β3|L)1/3≤ 0.324.

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Dispersion Limitations (cont.)

• Even a 1-nm spectral width limits BL < 0.1 (Gb/s)-km.

• DFB lasers essentialfor most lightwave systems.

• For B > 2.5 Gb/s, dispersion managementrequired.

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Dispersion Limitations (cont.)Effect of Frequency chirp

• Numerical simulations necessary for more realistic pulses.

• Super-Gaussian pulse shape often used: A(0,T)= A0exp

[−1+iC

2

(t

T0

)2m].

• Chirp can affect the system performance drastically.

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Fiber LossesDefinition: α(dB/km) =−10

L log10

(PoutPin

)≈ 4.343α .

• Material absorption (silica, impurities, dopants)

• Rayleigh scattering (varies as λ−4)

• Waveguide imperfections (macro and microbending)

DispersionConventional Fiber

Dry Fiber

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Major Nonlinear Effects• Stimulated Raman Scattering (SRS)

• Stimulated Brillouin Scattering (SBS)

• Self-Phase Modulation (SPM)

• Cross-Phase Modulation (XPM)

• Four-Wave Mixing (FWM)

Origin of Nonlinear Effects in Optical Fibers

• Ultrafast third-order susceptibility χ (3).

• Real part leads to SPM, XPM, and FWM.

• Imaginary part leads to SBS and SRS.

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Brillouin Scattering• Scattering of light from acoustic waves (electrostriction).

• Energy and momentum conservation laws require

ΩB = ωp−ωs and kA = kp−ks.

• Brillouin shift : ΩB = |kA|vA = 2vA|kp|sin(θ/2).

• only possibility θ = π for single-mode fibers (backward propagating

Stokes wave).

• Using kp = 2πn/λp, νB = ΩB/2π = 2nvA/λp.

• With vA = 5.96 km/s and n = 1.45, νB≈ 11 GHz near 1.55 µm.

• Stokes wave grows from noise.

• Not a very efficient process at low pump powers.

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Stimulated Brillouin Scattering• Becomes a stimulated process at high input power levels.

• Governed by two coupled equations:

dIpdz

=−gBIpIs−αpIp, −dIsdz

= +gBIpIs−αsIs.

• Brillouin gain has a narrow Lorentzian spectrum

(TB≈ 10 ns; ∆ν ∼ 20 MHz)

gB(Ω) =gB(ΩB)

1+(Ω−ΩB)2T2B

.

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SBS Threshold• Threshold condition: gBPthLeff/Aeff ≈ 21.

• Effective fiber length: Leff = [1−exp(−αL)]/α .

• Effective core area: Aeff ≈ 50–80 µm2.

• Peak Brillouin gain: gB≈ 5×10−11 m/W.

• Low threshold power for long fibers (∼5 mW).

• Most of the power reflected backward after SBS threshold.

Threshold can be increasedusing

• Phase modulation at frequencies >0.1 GHz.

• Sinusoidal strain along the fiber.

• Nonuniform core radius or dopant density.

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Stimulated Raman Scattering• Scattering of light from vibrating molecules.

• Scattered light shifted in frequency.

• Raman gain spectrum extends over 40 THz.

(a)

(b)

• Raman shift at Gain peak: ΩR = ωp−ωs∼ 13 THz).

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SRS Threshold• SRS governed by two coupled equations:

dIpdz

=−gRIpIs−αpIp,dIsdz

= gRIpIs−αsIs.

• Threshold condition: gRPthLeff/Aeff ≈ 16.

• Peak Raman gaingR≈ 6×10−14 m/W near 1.5 µm.

• Threshold power relatively large (∼ 0.6 W).

• SRS is not of concern for single-channel systems.

• Leads to interchannel crosstalk in WDM systems.

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Self-Phase Modulation• Refractive index depends on intensity as

n(ω, I) = nL(ω)+n2I(t).

• n2 = 2.6×10−20 m2/W for silica fibers.

• Propagation constant: β ′ = β +k0n2P/Aeff ≡ β + γP.

• Nonlinear parameter γ = 2πn2/(Aeffλ ).

• Leads to nonlinear Phase shift:

φNL =∫ L

0(β ′−β )dz=

∫ L

0γP(z)dz= γPinLeff.

• Optical field modifies its own phase (SPM).

• Phase shift varies with time for pulses (chirping).

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SPM-Induced Chirp

• Chirp depends on the pulse shape.

• Gaussian pulses (m= 1): Nearly linear chirp across the pulse.

• Super-Gaussian pulses (m= 1): Chirping only near pulse edges.

• In all cases, SPM leads to spectral broadening.

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Cross-Phase Modulation• Nonlinear refractive index seen by one wave depends on the

intensity of other copropagating channels.

• Nonlinear index for two channels:

∆nNL = n2(|A1|2+2|A2|2).

• Total nonlinear phase shift for multiple channels

φNLj = γLeff

(Pj +2 ∑

m6= j

Pm

).

• XPM induces a nonlinear coupling among channels.

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Nonlinear Schrodinger Equation• SPM and XPM effects can be included by adding a nonlinear term

to the equation used earlier for dispersive effects.

• This equation is known as the nonlinear Schrodinger equation:

∂A∂z

+iβ2

2∂ 2A∂ t2

= iγ|A|2A.

• Fibers with large Aeff help through reduced γ .

• New fiber: large effective-area fiber or LEAF .

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Four-Wave Mixing• FWM converts two photons from one or two pump beams into two

new frequency-shifted photons.

• Energy conservation: ω1+ω2 = ω3+ω4.

• Degenerate FWM: 2ω1 = ω3+ω4.

• Momentum conservation or phase matchingis required.

• FWM efficiency governed by phase mismatch:

∆ = β (ω3)+β (ω4)−β (ω1)−β (ω2).

• In the degenerate case (ω1 = ω2), ω3 = ω1+Ω, and ω4 = ω1−Ω.

• Expanding β in a Taylor series, ∆ = β2Ω2.

• FWM becomes important for dense WDM systems designed with

low β2 fibers.

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FWM: Good or Bad?• FWM leads to interchannel crosstalk in WDM systems.

• It can be avoided through dispersion management.

On the other hand . . .

FWM can be used beneficially for

• Parametric amplification

• Optical phase conjugation

• Demultiplexing of OTDM channels

• Wavelength conversion of WDM channels

• Supercontinuum generation

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Fiber Design

• Core doped with GeO2; cladding with fluorine.

• Index profile rectangular for standard fibers.

• Triangular index profile for dispersion-shifted fibers.

• Raised or depressed cladding for dispersion control.

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Commercial FibersCorning SMF-28 λZD = 1.31 µm, D (1.55 µm) = 17 ps/(km-nm),

S = 0.09 ps/(km-nm2), α = 0.21 dB/km,

Aeff = 60 µm2, Dp > 0.2 ps/√

km.

Lucent TrueWave RS λZD = 1.45 µm, D (1.55 µm)

= 4.4 ps/(km-nm), S = 0.04 ps/(km-nm2), α = 0.205 dB/km,

Aeff = 55 µm2, Dp < 0.05 ps/√

km.

Corning LEAF λZD = 1.50 µm, D (1.55 µm) = 4.2 ps/(km-nm),

S = 0.085 ps/(km-nm2), α = 0.21 dB/km,

Aeff = 72 µm2, Dp < 0.07 ps/√

km.

Lucent AllWave λZD = 1.31 µm, D (1.55 µm) = 17 ps/(km-nm),

S = 0.09 ps/(km-nm2), α = 0.19 dB/km,

Aeff = 80 µm2, Dp < 0.05 ps/√

km.

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Fabrication MethodsTwo-Stage Fabrication

• preform: Length 1 m, diameter 2 cm; correct index profile.

• Preform drawn into fiber using a draw tower.

Preform Fabrication Techniques

• Modified chemical vapor deposition (MCVD).

• Outside vapor deposition (OVD).

• Vapor Axial deposition (VAD).

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Fiber Draw Tower

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Fiber CablesLight-Duty Cables

Heavy-Duty Cables