Fiber Optical Communication Lecture 5, Slide 1

Lecture 5

• Why/when are nonlinear phenomena important?

• Different types of fiber nonlinearities

• The Kerr effect: SPM, XPM, FWM

• Impact from dispersion

• Solitons

Fiber Optical Communication Lecture 5, Slide 2

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

Fiber Optical Communication Lecture 5, Slide 3

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...

Fiber Optical Communication Lecture 5, Slide 4

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

Fiber Optical Communication Lecture 5, Slide 5

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)

Fiber Optical Communication Lecture 5, Slide 6

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

Fiber Optical Communication Lecture 5, Slide 7

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

Fiber Optical Communication Lecture 5, Slide 8

• 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

Fiber Optical Communication Lecture 5, Slide 9

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

Fiber Optical Communication Lecture 5, Slide 10

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

Fiber Optical Communication Lecture 5, Slide 11

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

Fiber Optical Communication Lecture 5, Slide 12

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)

Fiber Optical Communication Lecture 5, Slide 13

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)

Fiber Optical Communication Lecture 5, Slide 14

GVD and SPM, the NLSE• In the typical case, dispersion and nonlinearities need to be considered

– The NLSE must be used

– For the time being, we ignore the loss

• A nonlinear PDE that has a formal analytical solution

– The inverse scattering transform

– Gives very few explicit solutions

• Solitons are special solutions which form stable pulses

– Propagate with either no width change or in a periodic manner

– Bright solitons must have β2 < 0 (anomalous dispersion) to be formed

• Pulse shaped (sech-shaped), fairly similar to Gaussian pulses

– Dark solitons exist for normal dispersion, β2 > 0

• Intensity dip in a constant amplitude

AAit

Ai

z

A 2

2

2

2

2

Fiber Optical Communication Lecture 5, Slide 15

Anomalous vs normal dispersion

• In the typical communication system

– Dispersion is strong and broadens the pulses, must be compensated for

– Nonlinearities are weak, hard to compensate for

• The nonlinearity affects the pulses differently in anomalous/normal disp.

– A weak nonlinearity can improve system performance (anomalous case)

Simulations and experiments, 145 km, 10 Gbit/s, NRZ system

Fiber Optical Communication Lecture 5, Slide 16

Cross-phase modulation, four-wave mixing• When more than one wavelength is present, we can often consider these

as separate waves

– Interact via cross-phase modulation (XPM) and four-wave mixing (FWM)

• Consider the case with two wavelengths A = a exp(–iωat) + b exp(–iωbt)

– Insert into |A|2A-term to see what it looks like

• SPM, first and second term

– The wave modulates its own phase

• XPM, third and fourth term

– Each wave modulates the index of refraction, which causes a phase modulation of the other wave

• FWM, fifth and sixth term

– New frequency components are generated (at 2ωb–ωa and 2ωa–ωb)

titititititi baabbaba eabebabeaaebbebaea)2(2*)2(2*2222

22

Fiber Optical Communication Lecture 5, Slide 17

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

Fiber Optical Communication Lecture 5, Slide 18

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

Fiber Optical Communication Lecture 5, Slide 19

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)

Fiber Optical Communication Lecture 5, Slide 20

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

Fiber Optical Communication Lecture 5, Slide 21

Solitons

Examples where solitons occur:

• Plasma physics

• Nonlinear optics

• Nerve impulses

• Josephson junctions

• Vortices (Jupiter’s red spot)

• Conduction in polymers

The Scottish engineer John Scott Russell (1808–1882) recalls the first observation of a soliton, while horseback riding near a canal:

”…rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed…

… and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation”

Fiber Optical Communication Lecture 5, Slide 22

Solitons in optical fibers• In soliton, there is a balance between dispersive and nonlinear effects

• Discussed in a fiber context in 1973 [Hasegawa et al., Appl. Phys. Lett. 23, 142 (1973)]

• Experimentally confirmed in 1980 [Mollenauer et al., Phys. Rev. Lett. 45, 1095 (1980)]

– Needed a laser source above 1.3 μm

• Soliton transmission has been demonstrated over millions of km (longest of all communication systems!) and bit-rates up to 160 Gbit/s

[L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers”, Physical Review Letters 45, 1095 (1980)]

Inset (box): The spectrum and the pulse directly after the laser

Top row: The spectra for...

Bottom row: ...the fiber output as a function of power (Fig shows autocorrelation traces)fundamental

soliton

Fiber Optical Communication Lecture 5, Slide 23

The fundamental soliton (a solution to the NLSE) is

For any (anomalous) dispersion and pulse width, there is an amplitude that creates a soliton

The NLSE has periodic solutions soliton states, breathers or higher-order solitons, which arise for the initial condition

where N is an integer

First order soliton: 0.5 < N < 1.5

Second order: 1.5 < N < 2.5 etc

Solitons in optical fibers

z

Ti

T

t

Tztu

2

0

2

0

2

0

2

2expchse,

0

2

0

2 chse0,T

t

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intensity

time distance

xx eex

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For solitons, the nonlinear length equals the dispersion length

Fiber Optical Communication Lecture 5, Slide 24

Problems with soliton communication

• Today soliton transmission is (typically) not used and nonlinear effects are kept low by design

– Why? Solitons seems to ideal information carriers!

• The soliton balance condition depends on the power

– Loss will decrease the power...

– ...and disturb the balance...

– ...and the pulses will start to broaden

• The solitons interact

– There is no superposition anymore!

– Pulses can attract or repel each other

• How to do amplitude modulation?