Nonlinear Fiber Optics and its Applications in Optical Signal ...

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Nonlinear Fiber Optics and its Applications in Optical Signal Processing Govind P. Agrawal Institute of Optics University of Rochester Rochester, NY 14627

c

2007 G. P. Agrawal

JJ II J I Back Close

Outline • Introduction

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• Self-Phase Modulation • Cross-Phase Modulation • Four-Wave Mixing • Stimulated Raman Scattering • Stimulated Brillouin Scattering • Concluding Remarks


Introduction Fiber nonlinearities

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• Studied during the 1970s. • Ignored during the 1980s. • Feared during the 1990s. • Are being used in this decade. Objective: • Review of Nonlinear Effects in Optical Fibers.


Major Nonlinear Effects • Self-Phase Modulation (SPM)

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• Cross-Phase Modulation (XPM) • Four-Wave Mixing (FWM) • Stimulated Raman Scattering (SRS) • Stimulated Brillouin Scattering (SBS) Origin of Nonlinear Effects in Optical Fibers • Ultrafast third-order susceptibility χ (3). • Real part leads to SPM, XPM, and FWM. • Imaginary part leads to SBS and SRS.


Self-Phase Modulation • Refractive index depends on optical intensity as (Kerr effect)

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n(ω, I) = n0(ω) + n2I(t). • Frequency dependence leads to dispersion and pulse broadening. • Intensity dependence leads to nonlinear phase shift φNL(t) = (2π/λ )n2I(t)L. • An optical field modifies its own phase (thus, SPM). • Phase shift varies with time for pulses (chirping). • As a pulse propagates along the fiber, its spectrum changes because of SPM.


Nonlinear Phase Shift • Pulse propagation governed by Nonlinear Schr¨odinger Equation ∂ A β2 ∂ A − + γ|A|2A = 0. 2 ∂z 2 ∂t

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2

i

• Dispersive effects within the fiber included through β2. • Nonlinear effects included through γ = 2πn2/(λ Aeff). • If we ignore dispersive effects, solution can be written as A(L,t) = A(0,t) exp(iφNL), where φNL(t) = γL|A(0,t)|2. • Nonlinear phase shift depends on input pulse shape. • Maximum Phase shift: φmax = γP0L = L/LNL. • Nonlinear length: LNL = (γP0)−1 ∼ 1 km for P0 ∼ 1 W.


SPM-Induced Chirp 1

2

(a)

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(b)

0.8 Chirp, δωT0

Phase, φNL

1 0.6 0.4

0 −1

0.2 −2 0

−2

−1

0 1 Time, T/T0

2

−2

−1

0 1 Time, T/T0

2

• Super-Gaussian pulses: P(t) = P0 exp[−(t/T )2m]. • Gaussian pulses correspond to the choice m = 1. • Chirp is related to the phase derivative dφ /dt. • SPM creates new frequencies and leads to spectral broadening.


SPM-Induced Spectral Broadening • First observed in 1978 by Stolen and Lin.

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• 90-ps pulses transmitted through a 100-m-long fiber. • Spectra are labelled using φmax = γP0L. • Number M of spectral peaks: φmax = (M − 12 )π. • Output spectrum depends on shape and chirp of input pulses. • Even spectral compression can occur for suitably chirped pulses.


SPM-Induced Spectral Narrowing 1

1 (b)

C=0

0.8

Spectral Intensity

Spectral Intensity

(a)

0.6 0.4 0.2

−2 0 2 Normalized Frequency

0.6 0.4

0 −4

4

1


4

1 (c)

(d)

C = −10

0.8

Spectral Intensity

Spectral Intensity

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0.2

0 −4

0.6 0.4 0.2 0 −4

C = 10

0.8

C = −20

0.8 0.6 0.4 0.2


4

0 −4


4

• Chirped Gaussian pulses with A(0,t) = A0 exp[− 12 (1 + iC)(t/T0)2]. • If C < 0 initially, SPM produces spectral narrowing.


SPM: Good or Bad? • SPM-induced spectral broadening can degrade performance of a lightwave system.

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• Modulation instability often enhances system noise. On the positive side . . . • Modulation instability can be used to produce ultrashort pulses at high repetition rates. • SPM often used for fast optical switching (NOLM or MZI). • Formation of standard and dispersion-managed optical solitons. • Useful for all-optical regeneration of WDM channels. • Other applications (pulse compression, chirped-pulse amplification, passive mode-locking, etc.)


Modulation Instability Nonlinear Schr¨odinger Equation ∂ A β2 ∂ A 2 − + γ|A| A = 0. ∂z 2 ∂t 2

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2

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• CW solution unstable for anomalous dispersion (β2 < 0). • Useful for producing ultrashort pulse trains at tunable repetition rates [Tai et al., PRL 56, 135 (1986); APL 49, 236 (1986)].


Modulation Instability • A CW beam can be converted into a pulse train.

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• Two CW beams at slightly different wavelengths can initiate modulation instability and allow tuning of pulse repetition rate. • Repetition rate is governed by their wavelength difference. • Repetition rates ∼100 GHz realized by 1993 using DFB lasers (Chernikov et al., APL 63, 293, 1993).


Optical Solitons • Combination of SPM and anomalous GVD produces solitons. 13/44

• Solitons preserve their shape in spite of the dispersive and nonlinear effects occurring inside fibers. • Useful for optical communications systems.

• Dispersive and nonlinear effects balanced when LNL = LD. • Nonlinear length LNL = 1/(γP0);

Dispersion length

LD = T02/|β2|.

• Two lengths become equal if peak power and width of a pulse satisfy P0T02 = |β2|/γ.


Fundamental and Higher-Order Solitons • NLS equation: i ∂∂Az − β22 ∂∂t A2 + γ|A|2A = 0. 2

• Solution depends on a single parameter:

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N2 =

γP0 T02 |β2 | .

• Fundamental (N = 1) solitons preserve shape: √ A(z,t) = P0 sech(t/T0) exp(iz/2LD). • Higher-order solitons evolve in a periodic fashion.


Stability of Optical Solitons • Solitons are remarkably stable. 15/44

• Fundamental solitons can be excited with any pulse shape.

Gaussian pulse with N = 1. Pulse eventually acquires a ‘sech’ shape.

• Can be interpreted as temporal modes of a SPM-induced waveguide. • ∆n = n2I(t) larger near the pulse center. • Some pulse energy is lost through dispersive waves.


Cross-Phase Modulation • Consider two optical fields propagating simultaneously.

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• Nonlinear refractive index seen by one wave depends on the intensity of the other wave as ∆nNL = n2(|A1|2 + b|A2|2). • Total nonlinear phase shift: φNL = (2πL/λ )n2[I1(t) + bI2(t)]. • An optical beam modifies not only its own phase but also of other copropagating beams (XPM). • XPM induces nonlinear coupling among overlapping optical pulses.


XPM-Induced Chirp • Fiber dispersion affects the XPM considerably.

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• Pulses belonging to different WDM channels travel at different speeds. • XPM occurs only when pulses overlap. • Asymmetric XPM-induced chirp and spectral broadening.


XPM: Good or Bad? • XPM leads to interchannel crosstalk in WDM systems.

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• 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


XPM-Induced Crosstalk 19/44

• A CW probe propagated with 10-Gb/s pump channel. • Probe phase modulated through XPM. • Dispersion converts phase modulation into amplitude modulation. • Probe power after 130 (middle) and 320 km (top) exhibits large fluctuations (Hui et al., JLT, 1999).


XPM-Induced Pulse Compression 20/44

• An intense pump pulse is copropagated with the low-energy pulse requiring compression. • Pump produces XPM-induced chirp on the weak pulse. • Fiber dispersion compresses the pulse.


XPM-Induced Mode Locking 21/44

• Different nonlinear phase shifts for the two polarization components: nonlinear polarization rotation. φx − φy = (2πL/λ )n2[(Ix + bIy) − (Iy + bIx )]. • Pulse center and wings develop different polarizations. • Polarizing isolator clips the wings and shortens the pulse. • Can produce ∼100 fs pulses.


Synchronous Mode Locking 22/44

• 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).


Four-Wave Mixing (FWM) 23/44

• FWM is a nonlinear process that transfers energy from pumps to signal and idler waves. • FWM requires conservation of (notation: E = Re[Aei(β z−ωt)]) ? Energy

ω1 + ω2 = ω3 + ω4

? Momentum

β1 + β2 = β3 + β4

• Degenerate FWM: Single pump (ω1 = ω2).


Theory of Four-Wave Mixing • Third-order polarization: PNL = ε0 χ (3)...EEE (Kerr nonlinearity). 4

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1 E = xˆ ∑ Fj (x, y)A j (z,t) exp[i(β j z − ω jt)] + c.c. 2 j=1 • The four slowly varying amplitudes satisfy i in2ω1 h dA1 2 2 ∗ i∆kz = f11|A1| + 2 ∑ f1k |Ak | A1 + 2 f1234A2A3A4e dz c k6=1 i in2ω2 h dA2 2 2 ∗ i∆kz = f22|A2| + 2 ∑ f2k |Ak | A2 + 2 f2134A1A3A4e dz c k6=2 h i dA3 in2ω3 2 2 ∗ −i∆kz = f33|A3| + 2 ∑ f3k |Ak | A3 + 2 f3412A1A2A4e dz c k6=3 h i in2ω4 dA4 2 2 ∗ −i∆kz = f44|A4| + 2 ∑ f4k |Ak | A4 + 2 f4312A1A2A3e dz c k6=4


Simplified Scalar Theory • Linear phase mismatch: ∆k = β3 + β4 − β1 − β2.

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• Overlap integrals fi jkl ≈ fi j ≈ 1/Aeff in single-mode fibers. • Full problem quite complicated (4 coupled nonlinear equations) • Undepleted-pump approximation =⇒ two linear coupled equations: • Using A j = B j exp[2iγ(P1 + P2)z], the signal and idler satisfy: √ √ dB3 dB4 ∗ −iκz = 2iγ P1 P2B4e , = 2iγ P1 P2B∗3e−iκz. dz dz • Total phase mismatch: κ = β3 + β4 − β1 − β2 + γ(P1 + P2). • Nonlinear parameter: γ = n2ω0/(cAeff) ∼ 10 W /km. −1

• Signal power P3 and Idler power P4 are much smaller than pump powers P1 and P2 (Pn = |An|2 = |Bn|2).


General Solution • Signal and idler fields satisfy: √ dB3 = 2iγ P1 P2B∗4e−iκz, dz

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dB∗4 dz

√ = −2iγ P1 P2B3eiκz.

• General solution when both the signal and idler are present at z = 0: B3(z) = {B3(0)[cosh(gz) + + B∗4(z) = {B∗4(0)[cosh(gz) − −

(iκ/2g) sinh(gz)] √ (iγ/g) P1P2B∗4(0) sinh(gz)}e−iκz/2 (iκ/2g) sinh(gz)] √ (iγ/g) P1P2B3(0) sinh(gz)}eiκz/2

• If an idler is not launched at z = 0 (phase-insensitive amplification): B3(z) = B3(0)[cosh(gz) + (iκ/2g) sinh(gz)]e−iκz/2 √ ∗ B4(z) = B3(0)(−iγ/g) P1P2 sinh(gz)eiκz/2


Gain Spectrum • Signal amplification factor for a FOPA: P3(L, ω) κ 2(ω) G(ω) = = 1+ 1+ 2 sinh2[g(ω) L] . P3(0, ω) 4g (ω) p • Parametric gain: g(ω) = 4γ 2P1P2 − κ 2(ω)/4.

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• Wavelength conversion efficiency: P4(L, ω) κ 2(ω) sinh2[g(ω) L]. ηc(ω) = = 1+ 2 P3(0, ω) 4g (ω) • Best performance for perfect phase matching (κ = 0): G(ω) = cosh2[g(ω) L],

ηc(ω) = sinh2[g(ω) L].


Parametric Gain and Phase Matching In the case p of a single pump: g(ω) = (γP0)2 − κ 2(ω)/4.

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Phase mismatch κ = ∆k + 2γP0 Parametric gain maximum when ∆k = −2γP0.

• Linear mismatch: ∆k = β2Ω2 +β4Ω4/12+· · · , where Ω = ωs −ω p. • Phase matching realized by detuning pump wavelength from fiber’s ZDWL slightly such that β2 < 0. • In this case Ω = ωs − ω p = (2γP0/|β2|)1/2.


Highly Nondegenerate FWM • Some fibers can be designed such that β4 < 0.

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• If β4 < 0, phase matching is possible for β2 > 0. • Ω can be very large in this case. 60 γP = 10 km−1 0

Frequency Shift (THz)

50

40

30 β2 = 0.1 ps2/km

20 0.01

10 0 0 −5 10

−4

−3

10

10 4

|β4| (ps /km)

−2

10


FWM: Good or Bad? • FWM leads to interchannel crosstalk in WDM systems.

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• It generates additional noise and degrades system performance.

On the other hand . . . FWM can be used beneficially for • Optical amplification and wavelength conversion • Phase conjugation and dispersion compensation • Ultrafast optical switching and signal processing • Generation of correlated photon pairs


Parametric Amplification • FWM can be used to amplify a weak signal.

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• Pump power is transferred to signal through FWM. • Peak gain G p = 14 exp(2γP0L) can exceed 20 dB for P0 ∼ 0.5 W and L ∼ 1 km. • Parametric amplifiers can provide gain at any wavelength using suitable pumps. • Two pumps can be used to obtain 30–40 dB gain over a large bandwidth (>40 nm). • Such amplifiers are also useful for ultrafast signal processing. • They can be used for all-optical regeneration of bit streams.


Singleand Dual-Pump FOPAs Four-Wave MixingFour-Wave (FWM) Mixing (FW 32/44 Pump

Pump 1

Idler

Signal

λ3

Pump 2

Signal

λ1

• Pump close ZDWL

λ4

to

fiber’s

• Wide but nonuniform gain spectrum with a dip

λ1 λ3

Idler

λ0

λ4 λ2

• Pumps at opposite ends • Much more uniform gain • Lower pump powers (∼0.5 W)


Optical Phase Conjugation • FWM generates an idler wave during parametric amplification.

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• Its phase is complex conjugate of the signal field (A4 ∝ A∗3) because of spectral inversion. • Phase conjugation can be used for dispersion compensation by placing a parametric amplifier midway. • It can also reduce timing jitter in lightwave systems.


Stimulated Raman Scattering • Scattering of light from vibrating silica molecules. 34/44

• 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 µm region.


Raman Threshold • Raman threshold is defined as the input pump power at which Stokes power becomes equal to the pump power at the fiber output:

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Ps(L) = Pp(L) ≡ P0 exp(−α pL). • P0 = I0Aeff is the input pump power. • For αs ≈ α p, threshold condition becomes Ps0eff exp(gRP0Leff/Aeff) = P0, • Assuming a Lorentzian shape for the Raman-gain spectrum, Raman threshold is reached when (Smith, Appl. Opt. 11, 2489, 1972) gRPthLeff ≈ 16 Aeff

=⇒

Pth ≈

16Aeff . gRLeff


Estimates of Raman Threshold Telecommunication Fibers 36/44

• For long fibers, Leff = [1 − exp(−αL)]/α ≈ 1/α ≈ 20 km for α = 0.2 dB/km at 1.55 µm. • For telecom fibers, Aeff = 50–75 µm2. • 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 • Because of gain, Leff = [exp(gL) − 1]/g > L. • For fibers with a large core, Aeff ∼ 1000 µm2. • Pth exceeds 10 kW for short fibers (L < 10 m). • SRS may limit fiber lasers and amplifiers if L 10 m.


SRS: Good or Bad? • Raman gain introduces interchannel crosstalk in WDM systems.

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• 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 1300–1650 nm range. • EDFA bandwidth limited to ∼40 nm near 1550 nm. • Distributed nature of Raman amplification lowers noise. • Needed for opening new transmission bands in telecom systems.


Stimulated Brillouin Scattering • Scattering of light from acoustic waves. 38/44

• Becomes a stimulated process when input power exceeds a threshold level. • Low threshold power for long fibers (∼5 mW). Reflected

Transmitted

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


Brillouin Shift • Pump produces density variations through electrostriction, resulting in an index grating which generates Stokes wave through Bragg diffraction.

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• Energy and momentum conservation require: ΩB = ω p − ωs,

~kA =~k p −~ks.

• Acoustic waves satisfy the dispersion relation: ΩB = vA|~kA| ≈ 2vA|~k p| sin(θ /2). • In a single-mode fiber θ = 180◦, resulting in νB = ΩB/2π = 2n pvA/λ p ≈ 11 GHz, if we use vA = 5.96 km/s, n p = 1.45, and λ p = 1.55 µm.


Brillouin Gain Spectrum 40/44

• 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 (νB = 2n pvA/λ p).


Brillouin Threshold • Pump and Stokes evolve along the fiber as 41/44

−

dIs = gBIpIs − αIs, dz

dIp = −gBIpIs − αIp. dz

• Ignoring pump depletion, Ip(z) = I0 exp(−αz). • Solution of the Stokes equation: Is(L) = Is(0) exp(gBI0Leff − αL). • Brillouin threshold is obtained from gBPthLeff ≈ 21 =⇒ Aeff

Pth ≈

21Aeff . gBLeff

• Brillouin gain gB ≈ 5 × 10−11 m/W is nearly independent of the pump wavelength.


Estimates of Brillouin Threshold Telecommunication Fibers 42/44

• For long fibers, Leff = [1 − exp(−αL)]/α ≈ 1/α ≈ 20 km for α = 0.2 dB/km at 1.55 µm. • For telecom fibers, Aeff = 50–75 µm2. • Threshold power Pth ∼1 mW is relatively small. Yb-doped Fiber Lasers and Amplifiers • Because of gain, Leff = [exp(gL) − 1]/g > L. • Pth exceeds 20 W for a 1-m-long standard fibers. • Further increase occurs for large-core fibers; Pth ∼ 400 W when Aeff ∼ 1000 µm2. • SBS is the dominant limiting factor at power levels P0 > 1 kW.


Techniques for Controlling SBS • Pump-Phase modulation: Sinusoidal modulation at several frequencies >0.1 GHz or with a pseudorandom bit pattern.

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• Cross-phase modulation by launching a pseudorandom pulse train at a different wavelength. • Temperature gradient along the fiber: Changes in νB = 2n pvA/λ p through temperature dependence of n p. • Built-in strain along the fiber: Changes in νB through n p. • Nonuniform core radius and dopant density: mode index n p also depends on fiber design parameters (a and ∆). • Control of overlap between the optical and acoustic modes. • Use of Large-core fibers: A wider core reduces SBS threshold by enhancing Aeff.


Concluding Remarks • Optical fibers exhibit a variety of nonlinear effects.

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• 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 applications: optical switching, soliton formation, wavelength conversion, broadband amplification, channel demultiplexing, etc. • New kinds of fibers have been developed for enhancing nonlinear effects (microstrctured fibers with air holes). • Nonlinear effects in such fibers are finding new applications.
