Download - 1 Chapter 2 Propagation of Signals in Optical Fiber Introduction to Optical Networks.

1

Chapter 2

Propagation of Signals in Optical

Fiber

Introduction to Optical Networks

2

2.Propagation of Signals in Optical Fiber

Advantages• Low loss ~0.2dB/km at 1550nm• Enormous bandwidth at least 25THz• Light weight• Flexible• Immunity to interferences• Low cost

Disadvantages and Impairments• Difficult to handle • Chromatic dispersion• Nonlinear Effects

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2.1 Light Propagation in Optical Fiber

Cladding refractive index ≈1.45

core 8~10μm, 50μm, 62.5μm doped

2,SiO

2SiO

5

2.1.1Geometrical Optical Approach (Ray Theory)

This approach is only applicable to multimode fibers.

incident angle ( 入射角 ) refraction angle ( 折射角 ) reflection angle ( 反射角 )

Snell’s Law

1

2

1

1 1

:

:

:r

r

1 1 2 2

1 2

sin sin

, :

n n

n n refractive indices

6

=>Critical angle

When total internal reflection occurs.

let = air refractive index

= acceptance angle (total reflection will occur at core/cladding interface)

1 2 22 / 2f n n and when

1 ,c

1 2

1

,c

nSin

n

0

max0

7

(2.2)

max max0 0 1 1

2

1

sin sin

sin ,c

n n

n

n

max12c

max 21

1

max1 1 2

2max 21 2

1

max 2 20 0 1 2

2 21 2max 1

00

sin( )2

cos

sin 1

sin

sin

n

n

n n

n

n

n n n

n n

n

8

If Δ is small (less than 0.01)

For (multimode)

Numerical Aperture NA=

Because different modes have different lengths of paths, intermodal dispersion occurs.

1 2

1

n nDenote

n

2 21 2 1 2 1 2( )( )n n n n n n

1 2 1( )n n n

2 21 2 1

max 1 10

0

2

2sin

n n n

n

n

1 1.5n 0.01

0 1n max0 12

max0 0 1sin 2n n

9

Infermode dispersion will cause digital pulse spreading

Let L be the length of the fiber

The ray travels along the center of the core

The ray is incident at (slow ray)1 /fT Ln C

c1

max1coss

LnT

c

2max1 21

2 1

cosLn n

cn n

s fT T T 21 1

2

1 1 2

2

21

2

( )

Ln Ln

cn c

Ln n n

cn

Ln

cn

10

Assume that the bit rate = Bb/s

Bit duration

The capacity is measured by BL (ignore loss)

Foe example, if

1T

B

1

2

2

1

2 2

1

2

TT

B

Ln

cn B

221 12 2

n c cBL

n n

10.01, 1.5n (10 / )BL mb s km

11

For optimum graded-index fibers, δT is shorter than that in the step-index fibers, because the ray travels along the center slows down (n is larger) and the ray traveling longer paths travels faster (n is small)

12

The time difference is given by (For Optical graded-index profile)

and

(single mode )

If

Long haul systems use single-mode fibers

21

8

LnT

c

21

1

24

TBc

BLn

10.01, 1.5 8 ( / )n BL Gb s km 12

cBL

n

13

2.1.2 Wave Theory Approach

Maxwell’s equations D.1 D.2 D.3 D.4 : the charge density, : the current density : the electric flux density, : the magnetic flux density : the electric field, : the magnetic field

0

D

B

B

t

DH J

t

��

��

��

ρ J��

D��

B��

��

H��

14

Because the field are function of time and location in the space, we denote them by

and , where and t are position vector and time.

Assume the space is linear and time-invariant the Fourier transform of is

2.4

let be the induced electric polarization 2.5 : the permittivity of vacuum 2.6 : the magnetic polarization : the permeability of vacuum 注意有些書 Fourier transform 定義為

( , )E r t��

H( , )r t��

r

( , )E r t��

-

E =( , ) ( , )exp( )r w E r t iwt dt

��

2w fP��

0D E P ��

00( )B H M

��

0

M

��

-

-

-

E =

E =

( , ) ( , )exp( )

( , )exp( 2 )

( , ) ( , )exp( 2 )

r w E r t jwt dt

E r t j ft dt

r t E r w j ft dt

��

��

��

15

Locality of Response: and related to dispersion and nonlinearities

If the response to the applied electric field is local

depends only on

not on other values of

This property holds in the 0.5~2μm wavelength

Isotropy: The electromagnetic properties are the same for all directions in the medium

Birefringence: The refraction indexes along two different directions are different (lithium niobate, LiNbO , modulator, isolator, tunable filter)

P��

E��

P 1( )r��

E 1( )r��

E 1 1( ),r r r��

3

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Linearity ： (Convolution Integral) 2.7

: linear susceptibility

The Fourier transform of is

2.8

Where is the Fourier transform of

( is similar to the impulse response)

is function of frequency

=> Chromatic dispersion

' ' '0( , ) ( , ) ( , )P r t x r t t E r t dt

��

( , )x r t

( , )P r t��

0( , ) ( , ) ( , )P r w x r w E r w

( , )x r w

( , )E r t��

( , )x r t

( , )x r w

17

Homogeneity: A homogeneous medium has the same electromagnetic properties at all points

The core of a graded-index fiber is inhomogeneous

Losslessness ： No loss in the medium

At first we will only consider the core and cladding regions of the fiber are locally responsive, isotropic, linear, homogeneous, and lossless.

The refractive index is defined as 2.9For silica fibers

( , ) ( )x r t x t

2( ) 1 ( )n w x w def

1.25 1.5x n

18

From Appendix D

For (zero charge)

(zero conductivity, dielectric material)

For nonmagnetic material

0

D

B

B

t

DH J

t

��

��

��

0

0

0

0

0

( )

J E

D E P

B H M

��

��

��

0M ��

19

Assume linear and homogenence

E Bt

��

0

2

0 2

2

0 02

2 2

0 0 02 2

( )

( )

( )

Ht

D

t t

E Pt

E Pt t

��

��

��

��

( , ) ( , )exp( )

1( , ) ( , )exp( )

2

E r w E r t iwt dt

E r t E r t iwt dw

iwt

��

��

20

Take Fourier transform

Recall

2.8

Denote

c: speed of light (Locally response, isotropic, linear,

homogeneous, lossless) 2.9

2 2

0 0 02 2( , ) ( , ) ( , )E r t E r t P r t

t t

��

2 20 0 0( , ) ( , ) ( , )E r w w E r w w P r w

0( , ) ( , ) ( , )P r w x r w E r w

2 20 0 0 0( , ) ( , ) ( , ) ( , )E r w w E r w w x r w E r w

0 0

1c

20 0( , ) (1 ( , )) ( , )E r w w x r w E r w

( ) 1 ( )n w x w

2 2

2( , )

w nE r w

c

( )iwt

21

palacian operation

2.10

(free space wave number)

2

2 22

2

2

2 22

2

2 2 20

0

( , ) ( ( , )) ( , )

( , ) ( , ) ( ( , ))

( , ) 0,

( )( , ) ( , ) 0

( , ) ( ) ( , ) 0

2

E r w E r w E r w

w nE r w E r w E r w

c

E r t

w n wE r w E r w

c

or E r w n w K E r w

wwhere K c

��

22

For Cartesian coordinates

For Cylindrical coordinatesρ. φ and z

n:{ a: radius of the core

Similarly 2.11

Boundary conditions is finite

and continuity of field at ρ=a

References: G.P. Agrawal “Fiber-Optical Communication System” Chapter 2John Senior “Optical Fiber Communications, Principles and practice”John Gowar “Optical Communication Systems”注意有些書在 time domain 運算有些書在 frequency domain 運算

2 2 22

2 2 2x y z

2 2 22 2

02 2 2 2

1 10

Ez Ez Ez Ezn k Ez

z

1

2

n a

n a

2 2

22

( )( , ) ( , ) 0

w n wH r w H r w

c

0 E , 0,E

23

Fiber Modes

claddingcore

x

z

y

24

must satisfy 2.1

0, 2.11 and the boundary conditions.

let

Where are unit vectors

For the fundamental mode, the longitudinal component is

the propagation constant

, , ,core cladding core claddingE E H and H��

( , ) x y zx y zE r w E e E e E e

, ,x y ze e and e

2 ( , )exp( )

2 2

:

z eE J x y i z

fn nwnc c

25

: Bessel functions

The transverse components

For cylindrical symmetry of the fiber

In general, we can write

(Appendix E)

( , )J x y

( )x yE and E

2 ( , )exp( )x tE J x y i z

2 2( , ) ( , )tJ x y and J x y depend only on x y

( , ) 2 ( , )exp( ( ) ) ( , )E r w J x y i w z e x y

26

Where

The multimode fiber can support many modes. A single mode fiber only supports the fundamental mode.

Different modes have different β,

such that they propagate at different speeds.=>mode dispersion

(We can think of a “mode” as one possible path that a guided ray can take)

2 2( , ) ( , ) ( , )tJ x y J x y J x y

27

For a fiber with core and cladding , if a wave propagating purely in the core, then the propagation constant is

λ: free space wavelength The wave numberSimilarly if the wave propagating purely in the cladding,

then

The fiber modes propagate partly in the cladding and partly in the core,

soDefine the effective index

The speed of the wave in the fiber=

1n 2n

1 11 1

2wn n knc

2k

2 2kn

2 1kn kn effn

k

2 1effn n n

eff

cn

28

For a fiber with core radius a , the cutoff condition is

: normalized wave number

Recall

V↓ when a↓ and ↓△

For a single mode fiber, the typical values are a=4μm and △=0.003

2 21 2

22.405

def

V a n n

1 2

1

n n

n

2 1550V at nm

V

29

The light energy is distributed in the core and the cladding.

31

Since Δ is small, a significant portion of the light energy can propagate in the cladding, the modes are weakly guided.

The energy distribution of the core and the cladding depends on wavelength.

It causes waveguide dispersion (different from material dispersion)

( Appendix E )For longer wave, it has more energy in the

cladding and vice versa.

1 2

2 1

eff

eff

kn kn

n kn n n

32

A multimode fiber has a large value of V

The number of modes

For example a=25μm, Δ=0.005

V=28 at 0.8μm

Define the normalized propagation constant (or normalized effective index)

2

2V

is used to investigate the wave

propagation in fibers

2 22 2 222

2 2 2 2 2 21 2 1 2

211( ) (1.1428 0.9960 / )

( )

( )

defeff

z z

n nk nb

k n k n n n

b V V HE

H E

b V

mode

33

PolarizationTwo fundamental modes exist for all λ. Others only

exist for λ< λcutoff,

Linearly polarized field : Its direction is constant.

For the fundamental mode in a single-mode fiber

( , )

:

, :

x y zx y z

z

x y

E r t E e E e E e

E longitudinal component

E E transverse components

��

,x y zE E E

36

Fibers are not perfectly circularly symmetric. The two orthogonally polarized fundamental modes have different β

=>Polarization-mode dispersion (PMD)Differential group delay (DGD)Δτ=Δβ/w ～ typical value Δτ=0.5100 km => 50 psPractically PMD varies randomly along the fiber and

may be cancelled from an segment to another segment.

Empirically, Δτ ～ 0.1-1 Some elements such as isolators, circulators, filters

may have polarization-dependent loss (PDL).

ps km

ps km

37

2.2 Loss and Bandwidth

10

10

:

:

10 log

(10 log ) 4.343

:

0.8 ~ 1.6

LPout Pin e

L length of fiber

fiber loss in dB km

PoutdB

PindB e

Two main loss mechanisms material absorption

and Rayleigh scattering

The material absorption is negligible in m m

39

Recall Take the bandwidth over which the loss in dB/km

is with a factor of 2 of its minimum.80nm at 1.3μm, 180nm at 1.55μm

=>BW=35 THz

2

cf

40

Erbium-Doped Fiber Amplifiers (EDFA) operate in the c and L bands, Fiber Raman Amplifiers (FRA) operate in the S band.

All Wave fiber eliminates the absorption peaks due to water.

42

2.2.1 Bending lossA bend with r = 4cm, loss < 0.01dB r↓ loss↑

2.2.3 Chromatic DispersionDifferent spectral components travel at different velocities.

a. Material dispersion n(w)

b. Waveguide dispersion, different wavelengths have different energy distributions in core and cladding =>different β, kn2< β < kn1

11

2

22

2

2

2

1, :

: ( )

0

0,

0,

dB group velocitydw

d B group velocity dispersion GVD parameterdw

If zero dispersion

the dispersion is normal

the dispersion is anomalous

43

2.3.1 Chirped Gaussian PulsesChirped: frequency of the pulse changes wit

h time.

Cause of chirp: direct modulation, nonlinear effects, generated on purpose. (soliton)

44

Appendix E, or Govind P. Agrawal “ Fiber- Optic Communication Systems” 2nd Edition, John Wiley & Sons. Inc. PP47~51

A chirped Gaussian pulse at z=0 is given by

The instantaneous angular frequency

2

0 0

2

0

2

0

1

20

21

20 0

0

1

20

2

00

( )

cos2

cos ( )

( )2

ik t

T i t

t

T

t

T

G t R A e e

k tA e t

T

A e t

ktt t

T

00

0

( )d t kt

dt T

T Pulse width

45

k = The chirp factorDefine: The linearly chirped pulse: the instantaneous a

ngular frequency increases or decreases with time, (k=constant)

Note

Solve with the initial

condition (E.7)

We get

(E.8)

A(z,t) is also Gaussian pulse

0( ) , i tG t R A o t e 2

1 2 22

A A i Ao

z t t

2

0

1

20( , )

ik t

TA o t A e

2

0 0 122

0 20 2

2

1

20 2

(1 )( )( , ) exp

2 (1 )(1 )

1exp

2 (1 )z

A T ik t zA z t

T i z ikT i z ik

t zikA

T i z ik

46

Broadening of chirped Gaussian pulses

They have the same of broadening length.

Note

0 2 2

2 2

0 2 2

2 2

2 22

0 0 0

1

z

z

T T kz i z

T kz z

T kz z

T T T

2 , 0.4D Da b distance L c d distance L

47

In Fig 2.9, , it is true for standard fibers at 1.55μm

Let be the dispersion length

If , dispersion can be neglected

2 0 2

0

2D

TL

20

2

Dz L

Tz

22 0

0(2.13) 1zTz T T

20

2

0

1 0 ( )

2zD

TIf and k unchirped pulsez

Tz L

T

0

0

2.5 / 1.55 ( )

0.2 ( ) 18002

10 / , 0.05 115

( ) 600 2.5 /

D

D

D

For Gb s systems at m return to zero pulse

Tlet T ns half pulse duration L km

For Gb s T ns L km

For NRZ return to zero L km for Gb s

48

For kβ2 < 0

For β2 > 0, high frequency travels faster

=> the tail travels fasters => compression

=> make βk < 0, LD increases

2 2

2 22 2

0 0 0

1zT k z z

T T T

↓decreases increases ↑ for certain z

(Fig 2.10)

49

Note β2> 0 , k< 0 β2k<0

50

2.3.2 Controlling the Dispersion ProfileDef:

Chromatic dispersion parameter

D =

D = DM + Dw

The standard single mode fiber has small chromatic dispersion at 1.3 μm but large at 1.55 μm

222 /c in ps nm km

51

At 1.55μm loss is low, and EDFA is well developed. Dispersion becomes an issue

We have not much control over DM, but Dw can be controlled by carefully designed refractive index profile.

Dispersion shifted fibers, which have zero dispersion in 1.55μm band

53

2.4 Nonlinear Effects

For bit rate 2.5 Gb/s, power a few mw≦ Linear Assumption is valid

Nonlinear effect appears for high power or high bit rate 10 Gb/s and WDM systems≧

The first category relates to the interaction of lightwave with phonons (molecular vibrations)

- Rayleigh Scattering- Stimulated Brillouin Scattering (SBS)- Stimulated Raman Scattering (SRS)

54

The second category is due to the dependence of the refractive index on the intensity

- self-phase modulation (SPM)- four-wave mixing (FWM)

SBS and SRS transfer energy from short λ (pump) to long λ(stokes wave)

Scattering gain coefficient, g, is measured in meter/watt and Δf.

● SPM induces chirping

In a WDM system, variation of n depending on the intensity of all channels.

=>Yields Cross-phase modulation (CPM)

=>interchannel crosstalk

55

FWM, 1 2, , ... nf f f

, , ( ), . 2 ,i j k i j k i j i j kf f f f f f e g f f f f f

56

2.4.1 Effective Length and AreaThe nonlinear effect depends on fiber length and cross-section.

0

0 0

( )

( )

:

1

10.22 1.55

20

z

L

e z

e

L

e

e

P z P e

P L P z dz

when L effective length

eL

Typically dB km at m L

L km

(for long link)

57

In addition nonlinear effect intensity

2

22

4

2 2

1

( , )

( , )

( , ) :

~ 85 , , ~ 50

50,

(

e

r

r

ee

e e

A effective cross sectional area

F r rdrd

F r rdrd

F r Fundamental mode intensity

PI effective intensityA

SMF A m DSF A m

DSF n n

dispersion compensating fiber

由 power point 大

21)

,e

DCF n n

A nonlinear effect

及差最多更小更嚴重

58

2.4.1 Stimulated Brillouin Scattering (SBS)

The scattering interaction occurs with acoustic phonons over Δf =15 MHz, at 1.55μm, stokes and pump waves propagate in opposite directions.

If spacing > 20 MHz => no effects on different channels

Ps(0) Pp(L)

pumping

SBS

59

(

(2.16)

1-= (P.78)

(0)

,

) (0)

( ) (0) :

(0) ( )

( ) (0)

B p e

e

s B p s p

p zp p p

Lp p

g P L

ALs s

L

e

Lp p

Assuming I is small g I I I

dII I z I e

dz

P L P e L length

P P L e e

eL

P L P e

114 10

(2.14)

(2.15)

: ,

: ,

:

B

sB p s s

pB p s p

s s s e

p p p e

e

mg independent ofwdI

g I I IdzdI

g I I IdzI Intensity of stokes P I A

I Intensity of pump P I A

A effective area

60

2.4.3 Stimulated Roman Scattering (SRS) SRS will deplete short wave power and amplifier long wave.

61

2.4.4 Propagation in a Nonlinear MediumIn a nonlinear medium, Fourier Transfer is not applicable.

When the electrical field has only one component, we can write and as the scalar functions , and .

Appendix F, contains higher order terms

( , )E r t��

( , )P r t��

( , )E r t

( , )P r t

( , )P r t

(1)0 1 1 1

(2)0 1 2 1 2 1 2

(3)0 1 2 3 1 2 3 1 2 3

(1)

( )

( , ) ( , ) ( , )

( , ) ( , ) ( , )

( , , ) ( , ) ( , ) ( , ) ( .1)

( , ) :

(

t

t t

t t t

i

P r t x r t t E r t dt

x t t t t E r t E r t dt dt

x t t t t t t E r t E r t E r t dt dt dt F

x r t the linear susceptibility

x r

, ) :

( , ) ( , ) ( , )L NL

t higher order nonlinear susceptibilities

P r t P r t P r t

linear polarization nonlinear polarization

62

Because of symmetry , and

The nonlinear response occurs less than 100x10-15sec.

If the bit rate is less than 100 Gb/s, then

: the third-order nonlinear susceptibility independent of time

For simplicity, assume that the signals are monochromatic plane waves

is constant in the plane perpendicular to the dispersion of propagation

In WDM systems with n wavelengths at the angular frequencies

(2)( , ) 0x r t ( ) 0 4. 5...ix i

(3)0 1 2 3 1 2 3 1 2 3( , ) ( , , ) ( , ) ( , ) ( , ) ( .2)

t t t

NLP r t x t t t t t t E r t E r t E r t dt dt dt F

(3) (3)1 2 3 1 2 3

(3) 3 30

( , , ) ( ) ( ) ( )

( , ) ( , ) (2.19)NL

x t t t t t t x t t t t t t

P r t x E r t E

(3)x

1

( , ) ( , ) cos( )n

i i i ii

E r t E z t E w t z

��

0( , ) ( , ) cos( )iE r t E z t E w t z ��

E

1 2 1 2, ... .,( , ... )n nw w w 0

63

2.4.5 Self-phase Modulation (SPM)Because n is intensity – dependent

=>induces phase shift proportional to the intensity

=>creates chirping => pulse broadening

It is significant for high power systems.

Consider a single channel case

0

2

(2.20)

3 shorter wavelength, the last term is negligible3

=( ) (2.21)

0 0

(3) 3 30 0 0

(3) 30 0 0 0 0

0

(3)0 0 0

( , ) cos( )

( , ) cos ( )

3 1cos( ) cos(3 3 )

4 4

3( , ) cos( )

4

NL

NL

E z t E w t z

P r t x E w t z

x E w t z w t z

w

P r t x E E w t z

( , )E z t

64

Recall for linear medium

Now, we have to modify as

We get

=> Phase changes with intensity

(1)2( ) 1n w x 2( )n w

(1)2 (3) 23( ) 1

4n w x x E

(1) (3) 200

(1)2

(3) 200 2

(3)

(3) 200

31

4

1

31

4

3( ) (2.22)

8

wx x E

c

let n x

w nx E

c n

x is very small

wn x E

c n

propagation constant changes with 2E2E

65

,whose phase changes as , this phenomenon is referal as self- phase modulation (SPM)

The intensity of the electrical field

The intensity-dependent refractive index is

The nonlinear index coefficient

in silica fiberWe take for exampleBecause a pulse has its finite temporal extent=>The phase shift is different in different parts of the pulseThe leading edges have positive frequency shift The tailing edges have negative frequency shift => SPM causes positive chirping

0 0( , ) cos( )E z t E w t z 2zE

220

1

2wI cnE in

m

(2.23)( )n E n nI

(3)

0

28

2 3

8

2.2 3.4 10

n xcn n

mn w

2

83.2 10 mn w

66

2.4.6 SPM-induced chirp for Gaussian PulsesConsider an unchirped pulse with envelope

which has unit peak amplitude and

-width T0=1, and the peak power P0=1Define the nonlinear length as

If link length L≧ NL => nonlinear effect is severe

2

2(0, )U e

1e

02

e

NL

AL

nP

67

From Appendix E, (E.18)

After propagation L distance,The SPM-induced phase change is

The instantaneous frequency is given by

References: Appendix E, and (Arg97)

2

2

(0, )

( , ) (0, ) .18

(0, )

NL

NL

izUL

izL

U z U z e E

U z e e

2

'NL

L eL

2

2

0 0

2

2( ) , : .

2( ) (1 2 )

NL

SPMNL

Lw w e w central freg

L

and the chirp factor is

Lk e

L

68

(2.25)

1 <

2 2(1 2 )

0

0

0

2( )

1

0

2,

2

1.55 , 0.22

1 384

10 38

SPMNL

L

e

eSPM NL

NL

NL

NL

Lk e increases with L

L

eRecall L effective length

At the center of the pulse

Ak L

L nP

dBAt m kmFor P mw L km negligible

P mw L km signifi

cant

69

2.4.7 Cross-Phase ModulationIn WDM systems, the intensity-dependent nonlinear effects

(phase shift) are enhanced by other signals, this effect is referred to as cross-phase modulation (CPM)

Consider two channels

Recall

1 1 1 2 2 2( , ) cos( ) cos( )E r t E w t z E w t z

(3) 30

3(3)0 1 1 1 2 2 2

( , ) ( , ) (2.19)

cos( ) cos( )

NLP r t x E r t

x E w t z E w t z

70

2w1+w2, 2w2+w1, 3w1and 3w2 can be neglected2w1-w2, 2w2-w1, are part of FWM. Consider the w1 channel, the CPM term is

If E1=E2

Apparently CPM effect is twice of SPM.In practice, β1 and β2 are different => The pulses corresponding to individual channel

walk away from each other. => can not interact further => CPM is negligible for standard fibersNote for DSF, they travel at same velocity, CPM is

significant

0

3

4(3) 2 2

1 2 1 1 1( 2 ) cos( ) (2.27)x E E E w t z

SPM CPM

71

2.4.8 Four-Wave Mixing (FWM)

72

wi, wj ,wk (three waves) generate

wi ± wj ± wk (fourth wave)

For example, channel spacing Δw

w2 = w1 + Δw, w3 = w1 + 2Δw

w1- w2+ w3 = w2, 2w2-w1 = w3

73

Define

The degeneracy factor

The normalized Pijk(z,t) is given by

If we assume that the optical signals propagate as plane waves over Ae and distance L, then the power is

, ,ijk i j kw w w w i j k

3 ( . 2.30). . 6 ( . 2.33)

i j eqi j k i j eqd

(3)

0( , ) cos ( ) ( ) (2.36)4ijk ijk i j k i j k i j k

xP z t d E E E w w w t z

2(3)2

8ijk ijk

ijk i j ke eff

w d xP PP P L

A n c

(using Fig 2.15 and 2.36)

.i j k i j kP P and P are powers at w w w

74

For example

If another channel at

Then FWM will interfere the channel.

Practical FWM lacks of phase matching

=> No significant influence (in normal fibers)

about 20dB below

2

28

1 ,

50

, 6

3.0 10

20

9.5

1

i j k

e

i j ijk

ijk

i

P P P mw

A m

w w d

mn wL km

P w

P mw

ijkwijkw

75

2.4.9 New Optical Fiber Types

A. DSF is not suitable for WDM due to nonlinear effect.

To reduce nonlinear effect (different group

velocities lack phase matching)

=>to develop nonzero-dispersion fibers

(NZ-DSF)

a chromatic dispersion 1~6 ps/nm-km

or -1 ~ -6 ps/nm-km

NZ-DSF has most advantage of DSF (in c-band)

77

Large Effective Area Fiber (LEAF)

nonlinear effect for fix power1

eA

79

Positive and Negative Dispersion Fibers For Chromatic dispersion compensation