A definition of Fiber Optics - ttu.ee 3 - 2016.pdfIn the time domain the dispersed pulse is The...

Fiber Optical Communication Lecture 3, Slide 1

Lecture 3

• Dispersion in single-mode fibers

– Material dispersion

– Waveguide dispersion

• Limitations from dispersion

– Propagation equations

– Gaussian pulse broadening

– Bit-rate limitations

• Fiber losses


Dispersion, qualitatively

• Different wavelengths (frequency components) propagate differently

• A pulse has a certain spectral width and will broaden during propagation

The index of refraction as a function of wavelength

The dispersion in SMF (red) and different dispersion-shifted fibers


Each spectral component of a pulse has a specific group velocity

The group delay after a distance L is

The group velocity is related to the mode group index given by

Assuming that Δω is the spectral width, the pulse broadening is governed by

where β2 is known as the GVD parameter (unit is s2/m or ps2/km)

Group delay, group index, and GVD parameter (2.3.1)

22

2

Ld

dL

d

dTT

d

ndn

c

d

ndn

c

n

cv

g

g

d

dL

v

LT

g

d

ndnng


The dispersion parameterMeasuring the spectral width in units of wavelength (rather than frequency), we can write the broadening as

ΔT = D Δλ L,

where D [ps/(nm km)] is called the dispersion parameter

D is related to β2 and the effective mode index according to

The dispersion parameter has two contributions:

material dispersion, DM: The index of refraction of the fiber material depends on the frequency

waveguide dispersion, DW: The guided mode has a frequency dependence

2

2

22

1

2

2

2222

212,

2

d

nd

d

nd

vd

dc

d

dv

d

dcD

g

g


Material dispersion (2.3.2)The material dispersion is related to the dependence of the cladding material’s group index on the frequency

An approximate relation for the material dispersion in silica is

where DM is given in ps/(nm km)

d

dnD

g

M

2

2

2

MMD 01122


Waveguide dispersion (2.3.3)The waveguide dispersion arises from the modes’ dependence on frequency

n2g: the cladding group index

V: the normalized frequency

b: the normalized waveguide index

dV

Vbd

d

dn

dV

VbVd

n

nD

gg

W

2

2

2

2

2

2

2

2

V

22

1

2

2

2

1 anc

nnaV

21

2

nn

nnb


Total dispersionThe total dispersion D is the sum of the waveguide and material contributions

D = DW + DM

Note: DW increases the net zero dispersion wavelength

The zero-dispersion wavelength is denoted either λ0 or λZD

An estimate of the dispersion-limited bit-rate is

|D|B Δλ L < 1

where B is the bit-rate, Δλ the spectral width, and L the fiber length


The dispersion can have different signs in a standard single-mode fiber (SMF)

D > 0 for λ > 1.31 μm: “anomalous dispersion”, the group velocity of higher frequencies is higher than for lower frequencies

D < 0 for λ < 1.31 μm: “normal dispersion”, the group velocity of higher frequencies is lower than for lower frequency components

Pulses are affected differently by nonlinear effects in these two cases

Anomalous and normal dispersion


Different fiber types• The fiber parameters can be tailored to shift the λ0-wavelength from

≈1.3 μm to 1.55 μm, dispersion-shifted fiber (DSF)

• A fiber with small D over a wide spectral range (typically with two λ0-wavelengths), dispersion-flattened fiber (DFF)

• A short fiber with large normal dispersion can compensate the dispersion in a long SMF, dispersion compensating fibers (DCF)

Dispersion compensating fiber


Fibers in the labThis dispersion compensating module contains 4 km of DCF...

...and it compensates the dispersion in this 25 km roll of SMF


Index profiles of different fiber types• Standard single-mode fiber

(SMF)

• Dispersion-shifted fiber (DSF)

• Dispersion-flattened fiber (DFF)


Higher order dispersion (2.3.4)• Near the zero-dispersion wavelength D ≈ 0

– The variation of D with the wavelength must be accounted for

– We have used β2 = 0

• S [ps/(nm2 km)] is called the dispersion slope

– Typical value in SMF is 0.07 ps/(nm2 km)

3

32

2

2

d

dc

d

dDS


Basic propagation equationWe will now develop the theory for signal propagation in fibers

The electric field is written as

• The field is polarized in the x-direction

• F(x, y) describes the mode in the transverse directions

• A(z, t) is the complex field envelope

• β0 is the propagation constant corresponding to ω0

Only A(z, t) changes upon propagation (described in the Fourier domain)

Each spectral component of a pulse propagates differently

)exp(),(),(ˆRe),( 00 tizitzAyxFt xrE

dtitzAzA

ziziAzA

)exp(),(),(~

)(exp),0(~

),(~

0


The propagation constant• The propagation constant is in general complex

– α is the attenuation

– δnNL is a small nonlinear (= power dependent) change of the refractive index

• Dispersion arises from βL(ω)

– The frequency dependence of βNL and α is small

• We now expand βL(ω) in a Taylor series around ω = ω0 (Δω = ω – ω0)

1/vp 1/vg GVD(rel. to D) dispersion slope(related to S)

0

332210 ...,)(

6)(

2)()(

m

m

mLd

d

2/)()()(2/)()/)](()([)( 00 iicnn NLLNL


Substitute β with the Taylor expansion in the expression for the evolution

of A(z, ω), calculate ∂A/∂z, and write in time domain by using Δω↔ i ∂/∂t

The nonlinearity is quantified by using δnNL = n2I where n2 [m2/W] is a

measure of the strength of the nonlinearity, and I is the light intensity

βNL = γ|A|2, where γ = 2πn2/(λ0Aeff) is the nonlinear coefficient

Aeff is the effective mode area and |A|2 is normalized to represent the power

γ is typically 1–20 W–1 km–1

Basic propagation equation (2.4.1)

AAit

A

t

Ai

t

A

z

ANL

262 3

3

3

2

2

21


Basic propagation equation• Use a coordinate system that moves with the pulse group velocity!

– This is called retarded time, t’ = t – β1z

– We neglect β3 to get

• This is the nonlinear Schrödinger equation (NLSE)

– The primes are implicit

• The loss reduces the power ⇒ reduces the impact from the nonlinearity

• The average power of the signal during propagation in the fiber is

• Note: α is in m-1 while loss is often expressed in dB/km

AAAit

Ai

z

A

22

2

2

2

2

2/

2/

av

2

av )0(),(1

lim)(

T

T

z

TePdttzA

TzP


Chirped Gaussian pulses (2.4.2)• To study dispersion, we neglect nonlinearity and loss

• The formal solution is

• Note: Dispersion acts like an all-pass filter

• We study chirped Gaussian pulses

– A0 is the peak amplitude

– C is the chirp parameter

– T0 is the 1/e half width (power)

02 2

2

2

t

Ai

z

A

ziAzA 22

2exp),0(

~),(

~

2

021

0 )/)(1(exp),0( TtiCAtA

00

2/1

FWHM 665.1)2(ln2 TTT


• For a chirped pulse, the frequency of the pulse changes with time

– What does this mean???

• Study a CW (continuous wave)

– A is a constant

• Writing A exp(iβ0z – iω0t) = A exp(iφ), we see that ω0 = –∂φ/∂t

• We define the chirp frequency to be

– We allow φ to have a time dependence

– We get φ from the complex amplitude

• In this way, the chirp frequency can depend on time

– For the Gaussian pulse we get ωc = Ct/T02

Chirp frequency

)exp(),(),(ˆRe),( 00 tizitzAyxFt xrE

ttc /)(


A linearly chirped pulseFrequency increases with time Frequency decreases with time

ωc ωc

tt


Time-bandwidth productThe Fourier transform of the input Gaussian pulse is

The 1/e spectral half width (intensity) is

The product of the spectral and temporal widths is

If C = 0 then the pulses are chirp-free and said to be transform-limited

as they occupy the smallest possible spectral width

Using the full width at half maximum (FWHM), we get

)1(2exp

1

2),0(

~2

0

22/1

2

00

iC

T

iC

TAA

0

2

0 /1 TC2

00 1 CT

22

FWHMFWHM 144.012ln2

CCT


We introduce ξ = z/LD where the dispersion length LD = T02/|β2|

In the time domain the dispersed pulse is

The output width (1/e-intensity point) broadens as

Chirped Gaussian pulses (2.4.2)

A Gaussian pulse remains Gaussian during propagation

The chirp, C1(ξ), evolves as the pulse propagates

If (C β2) is negative, the pulse will initially be compressed

C

i

bT

tiC

b

AtA

ff1

arctan22

)1(exp),(

22

0

2

10

)(sign

)1()(

)1()(

2

2

1

2/122

s

CsCC

sCb f

2/12

2

0

2

2

2

0

2

0

1 1)(

)(

T

z

T

zC

T

zTzb f


Broadening of chirp-free Gaussian pulses

Short pulses broaden more quickly than longer pulses

(Compare with diffraction of beams)

2

2

0

2

2

11)(

T

z

L

zzb

D

f


Broadening of linearly chirped Gaussian pulses

For (C β2) < 0, pulses initially compress and reaches a minimum at

z = |C|/(1+C2)LD at which C1 = 0 and

Chirped pulses eventually broaden more quickly than unchirped pulses0

2

0min

1

1

1

C

TT


Chirped Gaussian pulses in the presence of β3

Higher order dispersion gives rise to oscillations and pulse shape changes

2

3

0

2

3

2

2

0

2

2

2

0

2

2

0

2

24

)1(

221

CLLLC2/00 T


Effect from source spectrum widthUsing a light source with a broad spectrum leads to strong dispersive broadening of the signal pulses

In practice, this only needs to be considered when the source spectral width approaches the symbol rate

For a Gaussian-shaped source spectrum with RMS-width σω and with Gaussian pulses, we have

where Vω = 2σωσ0

2

3

0

3222

2

2

0

22

2

2

0

2

2

0

2

24)1(

2)1(

21

LVC

LV

LCp

Vω << 1 when the source spectral width << the signal spectral width


If, as for an LED light source, Vω >> 1 we obtain approximately

A common criteria for the bit rate is that

For the Gaussian pulse, this means that 95% of the pulse energy remains within the bit slot

In the limit of large broadening

σλ is the source RMS width in wavelength units

Example: D = 17 ps/(km nm), σλ = 15 nm ⇒ (BL)max ≈ 1 (Gbit/s) km

Limitations on bit rate, incoherent source (2.4.3)

22

0

2

2

2

0

2 )()( DLL

)4/(14/ BTB

14 DBL


In the case of operation at λ = λZD, β2 = 0 we have

With the same condition on the pulse broadening, we obtain

The dispersion slope, S, will determine the bit rate-distance product

Example: D = 0, S = 0.08 ps/(km nm2), σλ = 15 nm ⇒ (BL)max ≈ 20 (Gbit/s) km

Limitations on bit rate, incoherent source

22

212

0

22

3212

0

2 )()( SLL

18 2 SBL


For most lasers Vω << 1 and can be neglected and the criteria become

Neglecting β3:

The output pulse width is minimized for a certain input pulse width giving

Example: β2 = 20 ps2/km → (B2L)max ≈ 3000 (Gbit/s)2 km

500 km @ 2.5 Gbit/s, 30 km @ 10 Gbit/s

If β2 = 0 (close to λ0):

For an optimal input pulse width, we get

Limitations on bit rate, coherent source (2.4.3)

22

0

2

02

2

0

2 )2/( DL

14 2 LB

22

0

22

03

2

0

2 2/)4/( DL

324.0)( 3/1

3 LB


Limitations on bit rate, summary

A coherent source improves the bit rate-distance product

Operation near the zero-dispersion wavelength also is beneficial…

...but may lead to problems with nonlinear signal distortion


Dispersion compensation

• Dispersion is a key limiting factor for an optical transmission system

• Several ways to compensate for the dispersion exist

– More about this in a later lecture...

• One way is to periodically insert fiber with opposite sign of D

– This is called dispersion-compensating fiber (DCF)

– Figure shows a system with both SMF and DCF

– The GVD parameters are β21 and β22

• Group-velocity dispersion is perfectly compensated when

β21l1 + β22l2 = 0, which is equivalent to D1l1 + D2l2 = 0

• GVD and PMD can also be compensated in digital signal processing (DSP)


Fiber losses (2.5)• Fiber have low loss but the loss grows exponentially with distance

– Approx. 20–25 dB loss over 100 km

– Optical receivers add noise...

– ...and the input power may be too low to obtain sufficient SNR

• The optical power in a fiber decreases exponentially with the propagation distance as Pout = Pin exp(–αz)

– α is the attenuation coefficient (unit m-1)

• Often, attenuation is given in dB/km and its relation to α is

• Typical value in SMF at 1550 nm αdB = 0.2 dB/km ⇒α = 0.046 km-1 = 1/(21.7 km)

343.410log

10

10log

log10log10

110dB

LL e

Le

L


Attenuation mechanisms• Material absorption

– Intrinsic absorption: In the SiO2 material

• Electronic transitions (UV absorption)

• Vibrational transitions (IR absorption)

– Extrinsic: Due to impurity atoms

• Metal and OH– ions, dopants

• Rayleigh scattering

– Occurs when waves scatter off small, randomly oriented particles

– (Makes the sky blue!)

– Proportional to λ-4

• Waveguide imperfections

– Core-cladding imperfections on > λ length scales (Mie scattering)

– Micro-bending (bending curvature λ)

– Macro-bending (negligible unless bending curvature < 1–5 mm)


Total attenuation• Minimum theoretical loss is 0.15 dB/km at 1550 nm

• Main contributions: Rayleigh scattering and IR absorption

• Left figure: Theoretical curves and measured loss for typical fiber

• Right figure: Loss for sophisticated fiber with negligible loss peak


Lecture

• Why/when are nonlinear phenomena important?

• Different types of fiber nonlinearities

• The Kerr effect: SPM, XPM, FWM


Nonlinear effects• When is a phenomenon ”nonlinear”?

– Superposition does not apply

– The phenomenon is changed by an amplitude (power) change

• Which is the same, e.g., doubling the amplitude is equivalent to a superposition of a pulse on itself

• In nonlinear optics, light cannot be viewed as a superposition of independently propagating spectral components

– Spectral components interact

– New frequencies can be generated, existing components can lose power

• IR light can become visible (green)

• Fibers nonlinearity is important for moderate powers because

– The fiber core is small, the electric field intensity is high

– A fiber is long, allowing nonlinear distortion to accumulate


Why study fiber nonlinearities?• What transmitted power would you choose in a fiber optic link?

– Laser output power is sufficient

– The energy cost is small (typical input power is 1 mW)

• The figure shows that the SNR is proportional to the input power

• Clearly, higher input power is always better!?!

– No, actually it is not...


Why study fiber nonlinearities?• What limits the launch power?

• Before 1990: Limited by laser output power to 1 mW

• After 1990: EDFAs enable power levels up to > 100 mW

– Performance is limited by fiber nonlinearities

The nonlinear trade-off:

• Low power: System is limited by noise

• High power: System is limited by nonlinearities

There exist an optimum launch power

A higher power is not always better!

BER for a system with-out nonlinearities

Nonlinear limitation

Noise limitation


Nonlinearities in fibersTwo types of important nonlinear effects in fibers:

• Electrostriction

– Intensity modulation in the fiber leads to pressure changes in the density of the medium, which leads to changes of the refractive index

– Responsible for Stimulated Brillouin Scattering (SBS)

• The Kerr effect

– The refractive index is changed in proportion to the optical intensity

– This gives rise to

• Self-phase modulation (SPM)

• Cross-phase modulation (XPM)

• Four-wave mixing (FWM)

• Modulation instability

• Solitons, which propagate without any change of the shape

– The delayed response of the Kerr effect gives rise to a nonlinear frequency downshift called Stimulated Raman Scattering (SRS)


Nonlinearities in fibers, scattering processesStimulated Brillouin scattering

• Occurs only in the backward direction

• Light will be backscattered and downshifted 10 GHz

– Remaining photon energy is absorbed as a vibration mode in the fiber

• Requires power levels 10 mW

Stimulated Raman scattering

• Occurs both in the forward and backward direction

• Appears over a wide spectral range (15 THz, 100 nm)

• Photons are downshifted in frequency

– Remaining photon energy is absorbed by the fiber

• Requires power levels of about 0.1–1 W


Nonlinearities in fibers, the Kerr effect• The Kerr effect means that the refractive index is intensity dependent

– The propagation constant becomes β(ω) = βlin (ω) + γ|A(t)|2

• The Kerr-effect gives rise to

– Self-phase modulation (SPM)

• Causes spectral broadening

• Can counteract anomalous dispersion

• Can give rise to soliton pulses

– Solitons do not broaden in time or frequency

– Cross-phase modulation (XPM)

• Causes frequency shift of other WDM channels

• Limits WDM systems performance

– Four-wave mixing (FWM)

• Causes power exchange between WDM channels

• Limits WDM system performance

The fundamental phenomenon is SPM

XPM and FWM appear when we “interpret” SPM in a WDM system


• Start from the NLSE and eliminate loss term by

– U is the normalized amplitude

• The NLSE for U(z, t) becomes

• The function p(z) varies periodically between 1 and exp(–αLA)

– LA is the amplifier spacing

• Neglecting the impact from dispersion, the NLSE is

– LNL = 1/(γ P0) is the nonlinear length

Self-phase modulation (2.6.2)),()(),( 0 tzUzpPtzA

UUzpPit

Ui

z

U 2

02

2

2 )(2

UUL

zpi

z

U

NL

2)(

The nonlinear length is the propagation distance over which the nonlinear effects become important


The solution to the NLSE without dispersion is

The signal phase is changed by the signal itself ⇒ self-phase modulation

We have introduced Leff and φNL

• φNL is the nonlinear phase shift

• Leff is the effective length

– The power decreases during propagation, the nonlinearity becomes weaker

– Therefore, the effective length is shorter than the physical length

We have

where NA is the number of amplified sections of fiber (often called “spans”)

Self-phase modulation

),(exp),0(/),0(exp),0(),( eff

2tLitULLtUitUtLU NLNL

//)exp(1)()(0 0

eff AAA

L L

A NLNdzzpNdzzpLA


SPM impact on pulses

• In the absence of dispersion, the pulse shape will not change

• SPM introduces chirp and continually broadens the spectrum

• The chirping depends on pulse shape

– Super-Gaussian different from Gaussian pulse

• Solid line: A Gaussian pulse

• Dashed line: A super-Gaussian pulse with m = 3

(Remember the chirp frequency from last lecture)

eff0effmax / LPLL NL

2eff ),0()( tUtL

L

tt

NL

NL


Spectral broadening from SPM• Figures show the spectra

for chirped Gaussian pulses affected by SPM

• Dispersion and loss are neglected

• In this numerical example φmax = 4.5 π

• Spectral broadening will continue if more SPM is introduced

• Chirp on the pulse will change the effect from SPM significantly

• When φmax is large, the spectral broadening is strong

• Dispersion will change this result!

– SPM and GVD acting simultaneously leads to nontrivial phenomena


Linear dispersive effects

In the time domain:

• Pulses broaden...

– ...and start to interfere

• A phase shift (chirp) will become an amplitude change

The length scale for dispersion is the dispersion length LD = T0

2/|β2|

In the frequency domain:

• The amplitude is not changed

• Quadratic phase modulation

Fig. shows spectrum for single pulse

L = 1.5LD

L = 0

time (bit slots)

|A|2

|A|2

L = 0

L > 0

frequency (normalized)

|A|2

|A|2 arg(A)

arg(A)


Nonlinear propagation, SPM

In the time domain:

• The amplitude is not changed

• A pulse-shaped phase shift is introduced

– Self-phase modulation

In the frequency domain:

• The spectrum is broadened

• Energy is conserved

– Notice: Different y-scales

The length scale for the nonlinearity is the nonlinear length LNL = 1/(γ P0)

L > 0

L = 0

time (bit slots)

L = 0

L > 0

frequency (normalized)

|A|2

|A|2

|A|2

|A|2 arg(A)

arg(A)


Cross-phase modulation• Consider (again) the case A = a exp(–iωat) + b exp(–iωbt), insert into the

NLSE, neglect FWM, and split into a coupled system of equations

• The group velocities are different

– This causes walk-off and limits the impact of XPM

• The wave at ωa “notices” the presence of the wave at ωb through the additional nonlinear term

– And vice versa

• XPM is stronger than SPM by a factor of two, but walk-off limits the impact from XPM, i.e., dispersion reduces XPM

• The equation system can be used only for waves well separated in freq.

bbait

bi

t

b

vz

b

abait

ai

t

a

vz

a

bg

ag

22

2

2

2

,

22

2

2

2

,

22

1

22

1


Cross-phase modulation in WDM systems• XPM on channel b from channel a gives b → b exp[i2γPa(t)z]

– This changes the absolute phase, but can also...

– ...introduce a chirp that shifts the pulse up or down in frequency

• Figure shows that the sign of the shift depends on the pulse position

– Blue, solid line is the a channel, affected by the red, dashed b channel

– Remember the chirp frequency, ωc = –∂φ(t)/∂t

• The frequency shift depends on the relative position of the pulses

• The frequency shift will, via dispersion, give rise to timing jitter

• Dispersive walk-off will decrease the impact of XPM

frequency upshift no frequency change frequency downshift


Four-wave mixing• The waves at three frequencies generate a fourth

– The frequencies can be different or some may be the same

– With N different frequencies, FWM will generate N2(N–1)/2 mixing products

• The strength of each mixing product depends on

– The degeneracy (how many terms that contribute)

– How close the process is to being phase matched

• Phase matching is strongly dependent on the dispersion

– FWM is strong for low dispersion, e.g., near the zero-dispersion wavelength

– At symbol rates > 10 Gbaud, FWM is weak

Figure: Non-degenerate FWMLeft: Measured FWMRight: Original and generated frequencies (dispersion not accounted for)


Four-wave mixing in WDM systems• Equal channel spacing ⇒ FWM

components overlap with the data channels

– FWM can be a problem

• Solution:

– Decrease the dispersion length to reduce phase matching

• SMF/DCF better than DSF

• Only SMF is even better

– DSP dispersion compensation

– Use unequal channel spacing

• Not compliant with standard frequency assignment (ITU grid)

• Increases optical bandwidth

Original signal

Equalspacing

Unequal spacing

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