nonlinear effects in optical fibers-govind p. agrawal

1/100     Back Close Nonlinear Effects in Optical Fibers Govind P. Agrawal Institute of Optics University of Rochester Rochester, NY 14627 c 2006 G. P. Agrawal 2/100     Back Close Outline • Introduction • Stimulated Raman Scattering • Stimulated Brillouin Scattering • Self-Phase Modulation • Cross-Phase Modulation • Four-Wave Mixing • Supercontinuum Generation • Concluding Remarks 3/100     Back Close Introduction Fiber nonlinearities • Studied during the 1970s. • Ignored during the 1980s. • Feared during the 1990s. • May be conquered in this decade. Objective: • Review of Nonlinear Effects in Optical Fibers. 4/100     Back Close 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 χ (3) . • Real part leads to SPM, XPM, and FWM. • Imaginary part leads to SBS and SRS. 5/100     Back Close 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 µm region. 6/100     Back Close SRS Dynamics • SRS proces s is governed by two coupled equations: dI p dz = −g R I p I s − α p I p , dI s dz = g R I p I s − α s I s . • If we neglect pump depletion (I s  I p ), pump power decays exponentially, and the Stokes beam satisfies dI s dz = g R I 0 e −α p z I s − α s I s . • This equation has the solution I s (L) = I s (0)exp(g R I 0 L eff −α s L), L eff = [1−exp(−α p L)]/α p . • SRS acts as an amplifier if pump wavelength is chosen suitably. 7/100     Back Close 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 mode into the fiber: P s (L) =  ∞ −∞ ¯ hω exp[g R (ω p − ω)I 0 L eff − α s L]dω. • Approximate solution (using the method of steepest descent): P s (L) = P eff s0 exp[g R (Ω R )I 0 L eff − α s L]. • Effective input power is given by P eff s0 = ¯ hω s B eff , B eff =  2π I 0 L eff  1/2     ∂ 2 g R ∂ ω 2     −1/2 ω=ω s . 8/100     Back Close 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: P s (L) = P p (L) ≡ P 0 exp(−α p L). • P 0 = I 0 A eff is the input pump power. • For α s ≈ α p , threshold condition becomes P eff s0 exp(g R P 0 L eff /A eff ) = P 0 , • Assuming a Lorentzian shape for the Raman-gain spectrum, Raman threshold is reached when (Smith, Appl. Opt. 11, 2489, 1972) g R P th L eff A eff ≈ 16 → P th ≈ 16A eff g R L eff . 9/100     Back Close Estimates of Raman Threshold Telecommunication Fibers • For long fibers, L eff = [1 − exp(−αL)]/α ≈ 1/α ≈ 20 km for α = 0.2 dB/km at 1.55 µm. • For telecom fibers, A eff = 50–75 µm 2 . • Threshold power P th ∼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 < 100 m), L eff = L. • For fibers with a larg e core, A eff ∼ 500 µm 2 . • P th can exceed 100 kW depending on fiber length. • SRS may limit fiber lasers and amplifiers if L  10 m. 10/100     Back Close 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 1300–1650 nm range. • Erbium-doped fiber amplifiers limited to ∼40 nm. • Distributed nature of amplification lowers noise. • Likely to open new transmission bands. [...]... γ|A|2A = 0 2 ∂z 2 ∂t • 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 the pulse shape through its power profile P(t) = |A(0,t)|2 Back Close SPM-Induced Chirp 23/100 Nonlinear phase shift Experimental... simultaneously 29/100 • Nonlinear refractive index seen by one wave depends on the intensity of the other wave as ∆nNL = n2(|A1|2 + b|A2|2) • 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 Back Close XPM-Induced Chirp • Fiber dispersion affects... depends on optical intensity as 21/100 n(ω, I) = n0(ω) + n2I(t) • Leads to nonlinear Phase shift φNL(t) = (2π/λ )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 Back Close Nonlinear Phase Shift • Pulse propagation governed by Nonlinear Schr¨dinger... Ultrafast optical switching • Demultiplexing of OTDM channels • Wavelength conversion of WDM channels Back Close XPM-Induced Mode Locking 32/100 • 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... belonging to different WDM channels travel at different speeds • XPM occurs only when pulses overlap • Asymmetric XPM-induced chirp and spectral broadening Back Close XPM: Good or Bad? • XPM leads to interchannel crosstalk in WDM systems 31/100 • 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. .. reached Back Close Brillouin Shift • Pump produces density variations through electrostriction, resulting in an index grating which generates Stokes wave through Bragg diffraction 13/100 • 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... Responsible for the formation of optical solitons Back Close Modulation Instability Nonlinear Schr¨dinger Equation o 25/100 ∂ A β2 ∂ 2A i − + γ|A|2A = 0 2 ∂z 2 ∂t • CW solution unstable for anomalous dispersion (β2 < 0) • Useful for producing ultrashort pulse trains Back Close Modulation Instability • A CW beam can be converted into a pulse train 26/100 • A weak modulation helps to reduce the power level and... gain • Multiple gratings may need to be used for long fibers • For short fibers, a long grating can be made all along its length Back Close Grating-Induced SBS Suppression 20/100 [Lee and Agrawal, Opt Exp 11, 3467 (2003)] • (a) 15-ns pulses, 2-kW peak power, 1-m-long grating with κL = 35 • (b) Fraction of pulse energy transmitted versus grating strength Back Close Self-Phase Modulation • Refractive index... as λ p • For silica fibers g p ≈ 5 × 10−11 m/W, TB = Γ−1 ≈ 5 ns, and B gain bandwidth < 50 MHz Back Close Brillouin Gain Spectrum 15/100 • 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... 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: Wider core reduces SBS threshold by enhancing Aeff Back Close Fiber Gratings for Controlling SBS • Fiber Bragg gratings can be employed . the input pump power at which Stokes power becomes equal to the pump power at the fiber output: P s (L) = P p (L) ≡ P 0 exp(−α p L). • P 0 = I 0 A eff is the input pump power. • For α s ≈ α p ,. 1/100     Back Close Nonlinear Effects in Optical Fibers Govind P. Agrawal Institute of Optics University of Rochester Rochester, NY 14627 c 2006 G. P. Agrawal 2/100     Back Close Outline • Introduction •. exp[g R (ω p − ω)I 0 L eff − α s L]dω. • Approximate solution (using the method of steepest descent): P s (L) = P eff s0 exp[g R (Ω R )I 0 L eff − α s L]. • Effective input power is given by P eff s0 = ¯ hω s B eff ,