Photonics West Tutorial -SPIE

8/10/2019 Photonics West Tutorial -SPIE

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Nonlinear Effects in Optical Fibers

Govind P. Agrawal

Institute of Optics

University of Rochester

Rochester, NY 14627

c2006 G. P. Agrawal
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Outline

Introduction Stimulated Raman Scattering

Stimulated Brillouin Scattering

Self-Phase Modulation

Cross-Phase Modulation

Four-Wave Mixing Supercontinuum Generation

Concluding Remarks


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Introduction

Fiber nonlinearities Studied during the 1970s.

Ignored during the 1980s.

Feared during the 1990s.

May be conquered in this decade.

Objective:

Review ofNonlinear Effects in Optical Fibers.


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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 (3)

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

Imaginary part leads to SBS and SRS.


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Stimulated Raman Scattering

Scattering of light from vibrating silica molecules.

Amorphous nature of silica turns vibrational state into a band.

Raman gain spectrum extends over 40 THz or so.

Raman gain is maximum near 13 THz.

Scattered light red-shifted by 100 nm in the 1.5 m region.
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SRS Dynamics

SRS process is governed by two coupled equations:

dIp

dz=gRIpIspIp,

dIs

dz=gRIpIssIs.

If we neglect pump depletion (IsIp), pump power decays

exponentially, and the Stokes beam satisfiesdIs

dz=gRI0e

pzIssIs.

This equation has the solution

Is(L) =Is(0) exp(gRI0LeffsL), Leff= [1exp(pL)]/p.

SRS acts as an amplifier if pump wavelength is chosen suitably.


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Raman Threshold Even in the absence of an input, Stokes beam can buildup if pump

power is large enough.

Spontaneous Raman scattering acts as the seed for this buildup.

Mathematically, the growth process is equivalent to injecting

one photon per modeinto the fiber:

Ps(L) =

hexp[gR(p)I0LeffsL]d.

Approximate solution (using the method of steepest descent):

Ps(L) =Peff

s0 exp[g

R(

R)I

0L

eff

sL].

Effective input power is given by

Peffs0 =hsBeff, Beff=

2

I0Leff

1/2 2gR2

1/2

=s

.


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Estimates of Raman ThresholdTelecommunication Fibers

For long fibers, Leff= [1 exp(L)]/ 1/ 20km for

=0.2dB/km at 1.55 m.

For telecom fibers, Aeff= 5075 m2.

Threshold power Pth 1 W is too large to be of concern. Interchannel crosstalk in WDM systems because of Raman gain.

Yb-doped Fiber Lasers and Amplifiers

For short fibers (L


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SRS: Good or Bad?

Raman gain introduces interchannel crosstalk in WDM systems.

Crosstalk can be reduced by lowering channel powers but it limits

the number of channels.

On the other hand. . .

Raman amplifiers are a boon for WDM systems.

Can be used in the entire 13001650 nm range.

Erbium-doped fiber amplifiers limited to 40 nm.

Distributed nature of amplification lowers noise.

Likely to open new transmission bands.


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Raman Amplifiers

Pumped in backward direction using diode lasers.

Multiple pumps used to produce wide bandwidth with a relatively

flat gain spectrum.

Help to realize longer transmission distances compared with erbium-

doped fiber amplifiers.


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Stimulated Brillouin Scattering Scattering of light from acoustic waves.

Becomes a stimulated process when input power exceeds a

threshold level.

Low threshold power for long fibers (5 mW).

Transmitted

Reflected

Most of the power reflected backward after SBS threshold is reached.


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Brillouin Shift Pump produces density variations throughelectrostriction, resulting

in an index grating which generates Stokes wave through Braggdiffraction.

Energy and momentum conservation require:

B=ps,

kA=

kp

ks.

Acoustic waves satisfy the dispersion relation:

B=vA|kA| 2vA|kp|sin(/2).

In a single-mode fiber =180, resulting in

B= B/2=2npvA/p 11 GHz,

if we use vA=5.96km/s, np=1.45, andp=1.55 m.


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Brillouin Gain Spectrum

Decay of acoustic waves as exp(Bt) leads to aLorentzian gain

spectrum of the form

gB() =gp(B/2)

2

(B)2 + (B/2)2.

Peak gain depends on the material parameters as

gp gB(B) = 822e

np2p0cvAB.

Electrostrictive constant e=0(d/d)=0 0.902for silica.

Gain bandwidth B scales with p as2p .

For silica fibers gp 51011 m/W, TB=

1B 5ns, and

gain bandwidth


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Brillouin Gain Spectrum

Measured spectra for (a) silica-core (b) depressed-cladding, and(c) dispersion-shifted fibers.

Brillouin gain spectrum is quite narrow (50 MHz).

Brillouin shift depends on GeO2 doping within the core.

Multiple peaks are due to the excitation of different acoustic modes.

Each acoustic mode propagates at a different velocity vA and thus

leads to a different Brillouin shift (B=2npvA/p).


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Brillouin Threshold Pump and Stokes evolve along the fiber as

dIs

dz=gBIpIsIs,

dIp

dz=gBIpIsIp.

Ignoring pump depletion, Ip(z) =I0 exp(z).

Solution of the Stokes equation:Is(L) =Is(0) exp(gBI0LeffL).

Brillouin threshold is obtained from

gBPthLeff

Aeff 21 Pth

21Aeff

gBLeff.

Brillouin gain gB 5 1011 m/W is nearly independent of the

pump wavelength.


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Estimates of Brillouin ThresholdTelecommunication Fibers

For long fibers, Leff= [1 exp(L)]/ 1/ 20km for

=0.2dB/km at 1.55 m.

For telecom fibers, Aeff= 5075 m2.

Threshold power Pth 1 mW is relatively small.Yb-doped Fiber Lasers and Amplifiers

For short fibers (L1 kW.


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Techniques for Controlling SBS Pump-Phase modulation: Sinusoidal modulation at several frequen-

cies >0.1 GHz or with a pseudorandom bit pattern.

Cross-phase modulation by launching a pseudorandom pulse train

at a different wavelength.

Temperature gradient along the fiber: Changes in B= 2npvA/pthrough temperature dependence ofnp.

Built-in strain along the fiber: Changes in B through np.

Nonuniform core radius and dopant density: mode index np also

depends on fiber design parameters (aand ).

Control of overlap between the optical and acoustic modes.

Use of Large-core fibers: Wider core reduces SBS threshold by en-

hancing Aeff.


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Fiber Gratings for Controlling SBS Fiber Bragg gratings can be employed for SBS suppression [Lee and

Agrawal, Opt. Exp. 11, 3467 (2003)].

One or more fiber grating are placed along the fiber, depending on

the fiber length.

Grating is designed such that it is transparent to the pump beam,

but Stokes spectrum falls entirely within its stop band.

Stokes is reflected by the grating and it begins to propagate in the

forward direction with the pump.

A new Stokes wave can still buildup, but its power is reduced be-

cause of the exponential nature of the SBS gain.

Multiple gratings may need to be used for long fibers.

For short fibers, a long grating can be made all along its length.


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Grating-Induced SBS Suppression

[Lee and Agrawal, Opt. Exp. 11, 3467 (2003)]

(a) 15-ns pulses, 2-kW peak power, 1-m-long grating with L= 35

(b) Fraction of pulse energy transmitted versus grating strength.


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Self-Phase Modulation

Refractive index depends onoptical intensityas

n(,I) =n0() +n2I(t).

Leads tononlinear Phase shift

NL(t) = (2/)n2I(t)L.

An optical field modifies its own phase (SPM).

Phase shift varies with time for pulses.

Each optical pulse becomes chirped.

As a pulse propagates along the fiber, its spectrum changes because

of SPM.


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Nonlinear Phase Shift

Pulse propagation governed by Nonlinear Schrodinger Equation

iA

z22

2A

t2+ |A|2A=0.

Dispersive effects within the fiber included through2.

Nonlinear effects included through =2n2/(Aeff).

If we ignore dispersive effects, solution can be written as

A(L, t) =A(0, t) exp(iNL), where NL(t) =L|A(0, t)|2.

Nonlinear phase shift depends on the pulse shapethrough its

power profile P(t) = |A(0, t)|2.


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SPM-Induced Chirp

Nonlinear phase shift Experimental Spectra

Pulse width = 90 ps, Fiber length = 100 m.
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SPM: Good or Bad?

SPM-induced spectral broadening can degrade performance of a

lightwave system.

Modulation instability often enhances system noise.

On the positive side . . .

Modulation instability can be used to produce ultrashort pulses athigh repetition rates.

SPM can be used for fast optical switching.

It has been used for passive mode locking.

Responsible for the formation of optical solitons.


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Modulation Instability

Nonlinear Schrodinger Equation

iA

z22

2A

t2+ |A|2A=0.

CW solution unstable for anomalous dispersion(2


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Modulation Instability

A CW beam can be converted into a pulse train.

A weak modulation helps to reduce the power level and makes the

repetition rate tunable.

Two CW beams at slightly different wavelengths can initiate

modulation instability.

Repetition rate governed by the wavelength difference.

Repetition rates 100 GHz realized using DFB lasers.
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Nonlinear Fiber-Loop Mirror

An example of the Sagnac interferometer.

Transmission through the fiber loop:

T=14f(1 f) cos2[(f 12

)P0L].

f= fraction of power in the CCW direction.

T=0 for a 3-dB coupler (loop acts as aperfect mirror)

Power-dependent transmission for f=0.5.


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Cross-Phase Modulation

Consider two optical fields propagating simultaneously.

Nonlinear refractive index seen by one wave depends on the inten-

sity of the other wave as

nNL=n2(|A1|2 +b|A2|

2).

Nonlinear phase shift:

NL= (2L/)n2[I1(t) +bI2(t)].

An optical beam modifies not only its own phase but also of other

copropagating beams (XPM).

XPM induces nonlinear couplingamong overlapping optical pulses.


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XPM-Induced Chirp

Fiber dispersion affects the XPM considerably.

Pulses belonging to different WDM channels travel at

different speeds.

XPM occurs only when pulses overlap.

Asymmetric XPM-induced chirp and spectral broadening.


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XPM: Good or Bad?

XPM leads to interchannel crosstalk in WDM systems.

It can produce amplitude and timing jitter.

On the other hand . . .

XPM can be used beneficially for

Nonlinear Pulse Compression

Passive mode locking

Ultrafast optical switching

Demultiplexing of OTDM channels

Wavelength conversion of WDM channels


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XPM-Induced Mode Locking

Different nonlinear phase shifts for the two polarization components:nonlinear polarization rotation.

xy= (2L/)n2[(Ix+bIy) (Iy+bIx)].

Pulse center and wings develop different polarizations. Polarizing isolator clips the wings and shortens the pulse.

Can produce100 fs pulses.


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Synchronous Mode Locking

Laser cavity contains the XPM fiber (few km long).

Pump pulses produce XPM-induced chirp periodically.

Pulse repetition rate set to a multiple of cavity mode spacing.

Situation equivalent to the FM mode-locking technique.

2-ps pulses generated for 100-ps pump pulses (Noske et al.,

Electron. Lett, 1993).


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XPM-Induced Switching

A MachZehnder or Sagnac interferometer can be used.

Output switched to a different port using a control signal that shifts

the phase through XPM.

If control signal is in the form of a pulse train, a CW signal can be

converted into a pulse train.

Ultrafast switching time (


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Four-Wave Mixing (FWM)

FWM is a nonlinear process that transfers energy ofpumps

tosignalandidlerwaves.

FWM requires conservation of (notation: E= Re[Aexp(izit)])

Energy 1+2=3+4

Momentum 1+2=3+4

Degenerate FWM: Single pump (1=2).


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FWM: Good or Bad?

FWM leads to interchannel crosstalk in WDM systems.

It generates additional noise and degrades system performance.

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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Parametric Amplification

FWM can be used to amplify a weak signal.

Pump power is transferred to signal through FWM.

Peak gain Gp= 14

exp(2P0L)can exceed 20 dB for

P0 0.5W and L 1km.

Parametric amplifiers can provide gain at any wavelengthusing

suitable pumps.

Two pumps can be used to obtain 3040 dB gain over

a large bandwidth (>40 nm).

Such amplifiers are also useful for ultrafast signal processing.


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Single- and Dual-Pump FOPAs

Pump

IdlerSignal

3 1 4

Pump close to fibers

ZDWL

Wide but nonuniform gain

spectrum with a dip

IdlerSignal

Pump 2Pump 1

1 3 0 4 2

Pumps at opposite ends

Much more uniform gain

Lower pump powers (0.5 W)


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Optical Phase Conjugation

FWM generates anidler waveduring parametric amplification.

Its phase is complex conjugate of the signal field (A4 A3) because

ofspectral inversion.

Phase conjugation can be used for dispersion compensation by plac-

ing a parametric amplifier midway.

It can also reduce timing jitter in lightwave systems.


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Highly Nonlinear Fibers

Silica nonlinearity is relatively weak (n2=2.61020 m2/W).

Applications of nonlinear effects require high input powers in com-

bination with long fiber lengths (>1 km).

Parameter=2n2/(Aeff)can be increased by reducing Aeff.

Such fibers are called highly nonlinear fibers. Examples includephotonic-crystal, tapered, and other microstructure fibers.


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Supercontinuum Generation

FWM in combination with SPM, XPM, and SRS can generate su-

perbroad spectrum extending over >200 nm.

Produced by launching short optical pulses into dispersion-

and nonlinearity-controlled fibers.

Photonic-crystal fiber Tapered fiber

Coen et al., JOSA B, Apr. 2002 Birk et al., OL, Oct. 2000


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Supercontinuum Applications

Potential applications include optical coherence tomography, carrier-

envelope phase locking, telecommunications, etc.

Spectral slicing can be used to produce 1000 or more channels

(Takara et al., EL, 2000).


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Concluding Remarks

Optical fibers exhibit a variety of nonlinear effects.

Fiber nonlinearities are feared by telecom system designers because

they can affect system performance adversely.

Fiber nonlinearities can be managed thorough proper system design.

Nonlinear effects are useful for many device and system applica-tions: optical switching, soliton formation, wavelength conversion,

broadband amplification, demultiplexing, etc.

New kinds of fibers have been developed for enhancing nonlinear

effects.

Supercontinuum generation in such fibers is likely to found new

applications.

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