1

Chapter 2. Optical Fibers

Geometrical Optics & Wave Optics in Fiber

Loss

Dispersion

Nonlinearity

2

2.1 Geometrical Optics & Wave Optics in Fiber

2.1.1 Geometrical Optics

suited for a >> λ Multimode Step-Index Optical Fiber

( 1%)

1 2n n ,

2 21 2 1 2

21 1

n -n n -nΔ=

2n n

weak-waveguided fiber

3

2.1.1 Geometrical Optics

0 i 1 r 1n sinθ =n sinθ =n cosφ

c 2 1Total internal reflection: sinφ n /n

0 i 1n sinθ =n 2Δ=NA (Numerical Aperture)

2 2 2 1/20 i 1 c 1 2n sinθ =n 1-sin φ =(n -n )

4

2.1.1 Geometrical Optics

Δ , NA , η (coupling efficiency between light source & fiber)

dispersion (multiplath dispersion)

c

Lthe longest path:

sinφ

the shortest path: L2

1 1

c 2

n nL Ltime delay: ΔT= ( L)= Δ

c sinφ c n－

max B

1ΔT T = , (bit slot)

B

-322

1

n cBL< , Δ , BL (n=1.5, Δ=2 10 , BL<100Mb/s km)

n Δ

5

2.1.1 Geometrical Optics Multimode Graded-Index Optical Fiber

α1

1 2

rn [1-Δ( ) ] r < a

n(r)= an (1-Δ)=n r a

α , SIOF

α=2 , parabolic-index

6

2.1.1 Geometrical Optics

2mπZ= , all rays recover their initial positions and directions

p1/21

2

2n ΔP= ( )

a

7

2.1.1 Geometrical Optics

21n ΔΔT

=L 8c

α = 2(1-Δ) ,

21

8cB L <

n Δ

8

2.1.2 Wave Optics

E , ΗMaxwell's Equations Wave Equation

cylindrical coordinatesGuided wave equation Helmholtz's Equation

boundaryconditions

tangenticalcomponents

general solution eigenvalue equation

1. Derivation

(x, y, z)

t

separate variables

numerical solving

mnβ

suited for a ~λ, wave theory

9

2.1.2 Wave Optics

2. Basic Concepts Normalize Frequency

Mode Index

01

2 20 1 2 1 0 01

21

TE2

V=k a n -n n 2 2.405 J (r) TM

HE

0

nk

模式在光纤中的分布光纤折射率分布

n2

n1

n2

n1

n2

>

10

2.1.2 Wave Optics Birefringence 双折射

HE11 → two orthogonally polarized fiber modes x, y

Degeneracy!

Uniform:

Nonuniform:x yn n birefringency PMD

L L

00 0= dz n k dz

linear → elliptical → linear 线偏 → 椭圆 → 线偏

modal birefringence m x yB n n

beat length Bm x y

LB n n

7

m BB ~10 , L ~10m

y

x

11

2.1.2 Wave Optics

12

2.1.2 Wave Optics

PMF: Polarization —Maintaining Fiber 4

m BB 10 , L ~10mm Small random birefringence fluctuations do not affect the light polarization significantly

Spot Size 2

x 2

rE A exp( )exp(i z)

w w: field radius

1.2<V<2.4 3/ 2 6w / a 0.65 1.619V 2.879V

Confinement Factor a 2 2

xcore 022

total x0

| E | r drP 2a1 exp( )

P w| E | r dr

V=2, Γ≈75%

V=1, Γ=20%

2<V<2.4

13

Chapter 2. Optical Fibers

Geometrical Optics & Wave Optics in Fiber

Loss

Dispersion

Nonlinearity

14

2.2 Fiber Loss

2.2.1 Attenuation Coefficient

Pin Pout

dPP, :attenuation coefficient

dz

integrating out inP P exp( L) (1/ km)

Lout10 10

in

10

P10 10log ( ) log (e )

4.343

L P L

10l

(dB / k

og e (1/

m)

km)

15

2.2.2 Loss Mechanism

1. Absorption (light energy → heat energy, 能量转移 ) Intrinsic absorption

Impurity absorption

Infrared (λ>7μm): vibrational resonances

ultraviolet (λ< 0.4μm): electronic resonances

0.8~1.6 μm : < 0.1dB/km

OH : 0.95, 1.24, 1.39 m

0.85 1.31 1.55 m

－ 吸收峰

， ， 吸收谷值OH

－ Dry fiber

All wave fiber

400nm Transitional metal (Fe, Cu, Co, Ni):

9101000dB / km eliminated

16

2.2.2 Loss Mechanism

2. Scattering ( 方向偏折， 能量不转移 ) Rayleigh Scattering → Intrinsic, on a scale smaller than λ

R R4

1, ,

4

1.31

1.55

1.55

1.31

Mie scattering → Waveguide Imperfection, on a scale larger than λ

3. Bending ( 模式泄漏，能量不转换 )

Macro － bending Installing

Micro － bending Cabling

17

课堂作业1. 某光纤通信线路长 80km ，使用的光纤芯层折射率为 1.45 ，

相对折射率差为 0.3 ％，截止波长为 1m ，光纤损耗只于瑞利散射有关。工作在 1310nm 时，输入功率为 1mW ，接收到功率为－ 28.5dBm 。请计算：（ 1 ）该光纤在 1550nm处的模场半径为多少？（ 2 ）光纤在 1550nm 处的损耗系数为多少 dB/km?

2. 如果模拟信号的带宽为 3.1kHz ，信噪比大于 30dB ，则转化数字信号后要求传输速率至少为：（ ）

A 、 31kb/s ； B 、 31Mb/s ； C 、 93kb/s ； D 、 91Mb/s 。

3. 请简要描述固定电话使用者是如何通过点对点光纤通信系统实现通话的？

18

Chapter 2. Optical Fibers

Geometrical Optics & Wave Optics in Fiber

Loss

Dispersion

Nonlinearity

19

Inter-symbol interference !!!

Bit 2 Bit 1 Bit 2 Bit 1 Bit 2 Bit 1

2.3 Dispersion in SMF

SMF

20

2.3 Dispersion in SMF

Chromatic Dispersion (CD)

Group Velocity Dispersion (GVD)

Intra-modal Dispersion

High-order Dispersion

Polarization Mode Dispersion (PMD)

gV

21

2.3.1 Group Velocity Dispersion

1. Dispersion parameter

L T= L /Vg

2

2 22

( / ) (1/ )( )

g gd L V d V d dT L L

d d d d

dL L T L

d

22 2

2 2 2, ( )

c c cT L

－

22

2.3.1 Group Velocity Dispersion

2 2

2( )

cD T L D

－ D: Dispersion Parameter, ps/(nm·km)

1B L D -1( / ) ( )g gd L V d V

T Ld d

2. D ~ λ

2

-1-1

2 2

2 2

2 2 2 2

1 2 1( ) ( )

( ) ( )

2 2 ( )

2 2 2

g g

g

d c dD

d V d V

d d dV n

d d d c

c d c d n dnD n

d c d c c d

c dn dn d n c dn d n

d d d d d

23

2.3.1 Group Velocity Dispersion

2

1 2

1

-

-

22

0

M W

n nb

n n

V n a D D D

d

d

2 2

2

2 22 2

2 22

2 1

2

g gM

g gW

dn dnD

d c d

n dnVd Vb d VbD

n dV d dV

n2g: the group index of the cladding material

24

2.3.2 Material Dispersion

Sellmeier equation:2M

j j22 2

j=1 j

j j

j j

M

β ωn (ω)=1+

ω -ω

ω : resonance frequency; β : oscillator strength

ω , β : fitting value according to experiments

pure silica fiber: λ=1.276μm, D =0

M 2

M 2

M ZD

λ<1.276μm, D <0, β >0,

λ>1.276μm, D >0, β <0,

1.25μm<λ<1.6μm,

D =122(1-λ /λ)

正常色散区，负色散

反常色散区，正色散

25

2.3.3 Waveguide Dispersion

~ ,wD a

26

2.3.3 Waveguide Dispersion

27

2.3.3 Dispersion Compensation

w 0, G.652, D>0 ~17ps/(nm km) @1.55 m

lossDispersion shifted Fiber: lowest @ m

dispersion

Dispersion Compensating Fiber: D<0 @1.55 m

D

－

① SMF ② DCF

D=16ps/(nm ·km)

L1=50km

D’=-100ps/(nm ·km)

L2=?

1 1

2 2 2

T 16 50

T ' 100

D L

D L L

①:

② : 1 2 2T T 8L km

Dispersion flattened Fiber: 1.3 ~1.6 m D － 很小

28

Dispersion Shifted Fiber

Refractive Index Profile of Optical Fiber

29

Dispersion Compensation

Chirped Bragg grating

• • • 

long short

G.652: Positive Dispersion@1550nm

30

2.3.4 High － Order Dispersion

ZD ZDλ=λ λ λ

2

2 3 22 2 3

m

m m

D = SΔλ2

2

B L Δλ D 1, D 0 B L

D 0, D 0, D ~ f(λ)

dD d 2πc 2πc 4πcS= = β = β + β

dλ dλ λ λ λ

d ββ =

dω

B L Δλ D 1 B L (Δλ) S 1

Δλ = 2nm

S=0.05ps/ nm km B L

λ =1.55μm

－

<5Tbit/s km

Really?

31

2.3.5 PMD

nx

nyEx

Ey

Pulse As It Enters the Fiber Spreaded Pulse As It Leaves the Fiber

32

1x 1y 1gx gy

1 x y 0 m 0

L LΔT L β β L β

V V

Δβ n n k B k

－

statistic modal birefringence

22 2 2T 1 C

C C

C

L Lσ = < ΔT > 2(Δβ ) L exp(- ) + -1

L L

L : correlation length

L: fiber length

33

2.3.5 PMD

C T 1 C p

p

P

L <<L, σ Δβ 2L L = D L

D : PMD parameter, ps/ km

G.652 D : 0.01 ~ 1ps/ km

A limiting factor for high bit rates lightwave systems

34

2.4 Dispersion － Introduced Limitations

Optical pulse: 2

00

1 ic tA(0, t) A exp

2 T

T0: half-width at 1/e intensity point C: frequency chirp

FWHM 0T 2 ln 2T (full-width at half-maximum)

35

2.4 Dispersion － Introduced Limitations

When the source spectrum is Gaussian with the RMS spectral width σw 22 22

2 2 2 32 22 2 2 3

0 0 0 0

LC L L1 (1 V ) (1 C V )

2 2 4 2

L: fiber length σ0: RMS width of the input Gaussian pulse

0V 2 2

20

: dispersion-introduced broadening of Gaussian input pulses

36

2.4 Dispersion － Introduced Limitations

1. Optical Sources with a Large Spectral Width

non-zero-dispersion wavelength 3neglecting

2 22 2 2 2 20 2 0 0 D

D

D 0 D

B

L DL

D L dispersion induced broadenin

T / 4, 95% energy of Gaussian pulse remains within a

g

,

14B 1 L

bit slo

D

t4

V 1, C 0

37

2.4 Dispersion － Introduced Limitations

zero-dispersion wavelength 2 0

2 22 2 2 2 20 3 0

2

1 1L SL

2 2

L S 1/ 8

2. Optical Sources with a Small Spectral Width V 1, C 0

non-zero-dispersion wavelength 3neglecting

22 2 2 20 2 0 0 D

1/ 2

D 0 2

2

L / 2

L / 2

1B L

4

38

2.4 Dispersion － Introduced Limitations

zero-dispersion wavelength 2 0

22 2 2 20 2 0 0 D

1/3

D 0 3

1/3

3

L / 4 / 2

L / 4

L 0.324

39

2.5 Nonlinearities in Optical Fiber

2.5.1 Stimulated light scattering

SRS (Stimulated Raman Scattering)

SBS (Stimulated Brillouin Scattering)

Rayleigh Scattering － not to generate new frequency

Raman Scattering － photon → stokes photon + optical phonon

能量大，频移大，单个原子振动 Brillouin Scattering － photon → stokes photon + acoustic phonon

能量小，频移小，多原子振动

both directions, frequency shift is large

backward directions, frequency shift is small

40

2.5.1 Stimulated light scattering

1. SRS

R th eff eff

R

th

eff eff

eff eff

3R

2

g P L / A 16

g : Raman gain coefficient

P threshold power

1 exp LL : effective interaction length L km

A effective core area A w

g 1 10 m / W

w 50 m

0.2dB / km

th

in

P 570 mW

P 10 mW,

SRS 影响小

41

Figure 2.18: (a) Raman gain spectrum of fused silica at λp = 1μm and (b) energy levels participating in the SRS process.

42

2.5.1 Stimulated light scattering

2. SBS

Note: frequency shift < 10GHz, channel spacing ~ 100GHz ,

B th eff eff

11B th

g P L / A 21

g 5 10 m / W P 10 mW

not to generate crosstalk, but the loss of signal power

43

Figure 2.17: Brillouin-gain spectra measured using a 1.525-μm pump for three fibers with different germanium doping: (a) silica-core fiber; (b) depressed-cladding fiber; (c) dispersion-shifted fiber. Vertical scale is arbitrary.

44

2.5.2 Phase ModulationSPM : Self Phase Modulation

XPM : Cross Phase Modulation

－－

Power dependence of refractive index

1. SPM nonlinear refraction

j j 2 eff

2

eff

n ' n n P / A j=1, 2. core, cladding respectively

n : nonlinear index coefficient

A :effective mod e area

－

propagation constant

0 2 eff 2 eff' k n P / A P, 2 n /(A )

45

2.5.2 Phase Modulation

1. SPM nonlinear phase shift

L L

NL in eff in0 0

NL in eff eff

in in1 1

' dL P(z)dz P L P(z) P exp( z)

11, P L L

0.2 dB / kmP / , P 22 mW

2 W km

46

2.5.2 Phase Modulation

2. XPM

MNL

j eff j mm j

NLj j j

L P 2 P

/ 2M 1 P , P 1mW

assuming equal channel power

SPM → distortion of optical pulse

XPM → intensity noise How to solve

Solution: effA Corning: Large Effective Area Fiber (LEAF)

47

2.5.3 Four-Wave Mixing

Also originates from the three-order nonlinear susceptibility

1 2 3 4

4 1 2 3

4 1 2 3

h h h h energy conservation

k k k k momentum conservation

��������������������������������������������������������

Easily to realize phase-matching at zero-dispersion wavelength !

1 24 1 2 3

4 1 2 3 4 1 3

4 1 2 3

h h h h2

k k k k

��������������������������������������������������������

DWDM: large effective area non-zero dispersion-shifted fiber

YOFC: LAPOSH

48

2.5.4 Evolvement of Fiber

1. G.651 MMF

2. G.652 SMF

3. G.653 DSF

4. G.655 NZDSF

6. Dry fiber, All-wave fiber

5. LEAF LAPOSH

7. DCF: D= － 100ps/(nm·km), 0.6 dB / km (Lucent)

RDCF: SMF, D’=16ps/(nm·km) reverse

C band: 1530 ~ 1570 nm S band: 1490 ~ 1530 nm

L band: 1570 ~ 1610 nm O band & E band: 1260 ~ 1490 nm

Problems: 2.5, 2.8, 2.11, 2.13, 2.16, 2.19

49

Fiber evolutionG.651

G.652

G.653

G.654

G.655

G.656

DCF

DCF Module

RDCF

LEAF

Dry fiber

C+L+S fiber

50

Cable

Optical fibers

Tube

Strain relief(e.g., Kevlar)Innerjacket

Sheath

Outerjacket

Connector