Nonlinear Fiber Optics and its Applications in Optical Signal Processing

Transcription

1 1/44 Nonlinear Fiber Optics and its Applications in Optical Signal Processing Govind P. Agrawal Institute of Optics University of Rochester Rochester, NY c 2007 G. P. Agrawal

2 Outline Introduction Self-Phase Modulation Cross-Phase Modulation Four-Wave Mixing Stimulated Raman Scattering Stimulated Brillouin Scattering Concluding Remarks 2/44

3 Introduction Fiber nonlinearities 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. 3/44

4 Major Nonlinear Effects Self-Phase Modulation (SPM) Cross-Phase Modulation (XPM) 4/44 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.

5 Self-Phase Modulation Refractive index depends on optical intensity as (Kerr effect) n(ω,i) = n 0 (ω) + n 2 I(t). 5/44 Frequency dependence leads to dispersion and pulse broadening. Intensity dependence leads to nonlinear phase shift φ NL (t) = (2π/λ)n 2 I(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.

6 Nonlinear Phase Shift Pulse propagation governed by Nonlinear Schrödinger Equation i A z β 2 2 A 2 t + 2 γ A 2 A = 0. 6/44 Dispersive effects within the fiber included through β 2. Nonlinear effects included through γ = 2πn 2 /(λa eff ). If we ignore dispersive effects, solution can be written as A(L,t) = A(0,t)exp(iφ NL ), where φ NL (t) = γl A(0,t) 2. Nonlinear phase shift depends on input pulse shape. Maximum Phase shift: φ max = γp 0 L = L/L NL. Nonlinear length: L NL = (γp 0 ) 1 1 km for P 0 1 W.

7 SPM-Induced Chirp Phase, φ NL 1 (a) 2 (b) Time, T/T 0 Chirp, δωt Time, T/T 0 Super-Gaussian pulses: P(t) = P 0 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. 7/44

8 SPM-Induced Spectral Broadening First observed in 1978 by Stolen and Lin. 8/44 90-ps pulses transmitted through a 100-m-long fiber. Spectra are labelled using φ max = γp 0 L. Number M of spectral peaks: φ max = (M 1 2 )π. Output spectrum depends on shape and chirp of input pulses. Even spectral compression can occur for suitably chirped pulses.

9 SPM-Induced Spectral Narrowing Spectral Intensity (a) 1 (b) C = 0 C = Spectral Intensity 9/ Normalized Frequency Normalized Frequency Spectral Intensity (c) C = 10 Spectral Intensity (d) C = Normalized Frequency Normalized Frequency Chirped Gaussian pulses with A(0,t) = A 0 exp[ 1 2 (1 + ic)(t/t 0) 2 ]. If C < 0 initially, SPM produces spectral narrowing.

10 SPM: Good or Bad? SPM-induced spectral broadening can degrade performance of a lightwave system. Modulation instability often enhances system noise. 10/44 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.)

11 Modulation Instability Nonlinear Schrödinger Equation i A z β 2 2 A 2 t + 2 γ A 2 A = 0. 11/44 CW solution unstable for anomalous dispersion (β 2 < 0). Useful for producing ultrashort pulse trains at tunable repetition rates [Tai et al., PRL 56, 135 (1986); APL 49, 236 (1986)].

12 Modulation Instability A CW beam can be converted into a pulse train. Two CW beams at slightly different wavelengths can initiate modulation instability and allow tuning of pulse repetition rate. 12/44 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).

13 Optical Solitons Combination of SPM and anomalous GVD produces solitons. Solitons preserve their shape in spite of the dispersive and nonlinear effects occurring inside fibers. Useful for optical communications systems. 13/44 Dispersive and nonlinear effects balanced when L NL = L D. Nonlinear length L NL = 1/(γP 0 ); Dispersion length L D = T 2 0 / β 2. Two lengths become equal if peak power and width of a pulse satisfy P 0 T 2 0 = β 2 /γ.

14 Fundamental and Higher-Order Solitons NLS equation: i A z β A t 2 + γ A 2 A = 0. Solution depends on a single parameter: N 2 = γp 0T 2 0 β 2. Fundamental (N = 1) solitons preserve shape: 14/44 A(z,t) = P 0 sech(t/t 0 )exp(iz/2l D ). Higher-order solitons evolve in a periodic fashion.

15 Stability of Optical Solitons Solitons are remarkably stable. Fundamental solitons can be excited with any pulse shape. 15/44 Gaussian pulse with N = 1. Pulse eventually acquires a sech shape. Can be interpreted as temporal modes of a SPM-induced waveguide. n = n 2 I(t) larger near the pulse center. Some pulse energy is lost through dispersive waves.

16 Cross-Phase Modulation Consider two optical fields propagating simultaneously. Nonlinear refractive index seen by one wave depends on the intensity of the other wave as 16/44 Total nonlinear phase shift: n NL = n 2 ( A b A 2 2 ). φ NL = (2πL/λ)n 2 [I 1 (t) + bi 2 (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.

17 XPM-Induced Chirp Fiber dispersion affects the XPM considerably. Pulses belonging to different WDM channels travel at different speeds. 17/44 XPM occurs only when pulses overlap. Asymmetric XPM-induced chirp and spectral broadening.

18 XPM: Good or Bad? XPM leads to interchannel crosstalk in WDM systems. It can produce amplitude and timing jitter. 18/44 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

19 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).

20 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.

21 XPM-Induced Mode Locking 21/44 Different nonlinear phase shifts for the two polarization components: nonlinear polarization rotation. φ x φ y = (2πL/λ)n 2 [(I x + bi y ) (I y + bi x )]. Pulse center and wings develop different polarizations. Polarizing isolator clips the wings and shortens the pulse. Can produce 100 fs pulses.

22 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).

23 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[Ae i(βz ωt) ]) Energy ω 1 + ω 2 = ω 3 + ω 4 Momentum β 1 + β 2 = β 3 + β 4 Degenerate FWM: Single pump (ω 1 = ω 2 ).

24 Theory of Four-Wave Mixing Third-order polarization: P NL = ε 0 χ (3).EEE (Kerr nonlinearity). E = 1 2 ˆx 4 j=1f j (x,y)a j (z,t)exp[i(β j z ω j t)] + c.c. 24/44 The four slowly varying amplitudes satisfy da 1 = in [( 2ω 1 dz c da 2 dz da 3 dz da 4 dz = in 2ω 2 c = in 2ω 3 c = in 2ω 4 c [( [( [( ) ] f 11 A f 1k A k 2 A f 1234 A 2A 3 A 4 e i kz k 1 ) ] f 22 A f 2k A k 2 A f 2134 A 1A 3 A 4 e i kz k 2 ) ] f 33 A f 3k A k 2 A f 3412 A 1 A 2 A 4e i kz k 3 ) ] f 44 A f 4k A k 2 A f 4312 A 1 A 2 A 3e i kz k 4

25 Simplified Scalar Theory Linear phase mismatch: k = β 3 + β 4 β 1 β 2. Overlap integrals f i jkl f i j 1/A eff 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(p 1 + P 2 )z], the signal and idler satisfy: db 3 dz = 2iγ P 1 P 2 B 4e iκz, db 4 dz = 2iγ P 1 P 2 B 3e iκz. Total phase mismatch: κ = β 3 + β 4 β 1 β 2 + γ(p 1 + P 2 ). Nonlinear parameter: γ = n 2 ω 0 /(ca eff ) 10 W 1 /km. Signal power P 3 and Idler power P 4 are much smaller than pump powers P 1 and P 2 (P n = A n 2 = B n 2 ). 25/44

26 General Solution Signal and idler fields satisfy: 26/44 db 3 dz = 2iγ P 1 P 2 B 4e iκz, db 4 dz = 2iγ P 1 P 2 B 3 e iκz. General solution when both the signal and idler are present at z = 0: B 3 (z) = {B 3 (0)[cosh(gz) + (iκ/2g)sinh(gz)] + (iγ/g) P 1 P 2 B 4(0)sinh(gz)}e iκz/2 B 4(z) = {B 4(0)[cosh(gz) (iκ/2g)sinh(gz)] (iγ/g) P 1 P 2 B 3 (0)sinh(gz)}e iκz/2 If an idler is not launched at z = 0 (phase-insensitive amplification): B 3 (z) = B 3 (0)[cosh(gz) + (iκ/2g)sinh(gz)]e iκz/2 B 4 (z) = B 3(0)( iγ/g) P 1 P 2 sinh(gz)e iκz/2

27 Gain Spectrum Signal amplification factor for a FOPA: [ 1 + G(ω) = P 3(L,ω) P 3 (0,ω) = ( 1 + κ2 (ω) 4g 2 (ω) ) ] sinh 2 [g(ω)l]. 27/44 Parametric gain: g(ω) = 4γ 2 P 1 P 2 κ 2 (ω)/4. Wavelength conversion efficiency: η c (ω) = P 4(L,ω) P 3 (0,ω) = ( 1 + κ2 (ω) 4g 2 (ω) ) sinh 2 [g(ω)l]. Best performance for perfect phase matching (κ = 0): G(ω) = cosh 2 [g(ω)l], η c (ω) = sinh 2 [g(ω)l].

28 Parametric Gain and Phase Matching In the case of a single pump: g(ω) = (γp 0 ) 2 κ 2 (ω)/4. 28/44 Phase mismatch κ = k + 2γP 0 Parametric gain maximum when k = 2γP 0. Linear mismatch: k = β 2 Ω 2 +β 4 Ω 4 /12+, where Ω = ω s ω p. Phase matching realized by detuning pump wavelength from fiber s ZDWL slightly such that β 2 < 0. In this case Ω = ω s ω p = (2γP 0 / β 2 ) 1/2.

29 Highly Nondegenerate FWM Some fibers can be designed such that β 4 < 0. If β 4 < 0, phase matching is possible for β 2 > 0. 29/44 Ω can be very large in this case γp 0 = 10 km 1 Frequency Shift (THz) β 2 = 0.1 ps 2 /km β 4 (ps 4 /km)

30 FWM: Good or Bad? FWM leads to interchannel crosstalk in WDM systems. It generates additional noise and degrades system performance. 30/44 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

31 Parametric Amplification FWM can be used to amplify a weak signal. Pump power is transferred to signal through FWM. 31/44 Peak gain G p = 1 4 exp(2γp 0L) can exceed 20 db for P W and L 1 km. Parametric amplifiers can provide gain at any wavelength using suitable pumps. Two pumps can be used to obtain 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.

32 Four-Wave Mixing Four-Wave (FWM) Mixing (FW Single- and Dual-Pump FOPAs Pump 32/44 Pump 1 Pump 2 Signal Idler Signal Idler λ 3 λ 1 λ 4 λ 1 λ 3 λ 0 λ 4 λ 2 Pump close to fiber s ZDWL Wide but nonuniform gain spectrum with a dip Pumps at opposite ends Much more uniform gain Lower pump powers ( 0.5 W)

33 Optical Phase Conjugation FWM generates an idler wave during parametric amplification. Its phase is complex conjugate of the signal field (A 4 A 3 ) because of spectral inversion. 33/44 Phase conjugation can be used for dispersion compensation by placing a parametric amplifier midway. It can also reduce timing jitter in lightwave systems.

34 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. 34/44 Raman gain is maximum near 13 THz. Scattered light red-shifted by 100 nm in the 1.5 µm region.

35 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: 35/44 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.

36 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 = 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 Because of gain, L eff = [exp(gl) 1]/g > L. For fibers with a large core, A eff 1000 µm 2. P th exceeds 10 kw for short fibers (L < 10 m). SRS may limit fiber lasers and amplifiers if L 10 m. 36/44

37 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. 37/44 On the other hand... Raman amplifiers are a boon for WDM systems. Can be used in the entire 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.

38 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). 38/44 Reflected Transmitted Most of the power reflected backward after SBS threshold is reached.

39 Brillouin Shift Pump produces density variations through electrostriction, resulting in an index grating which generates Stokes wave through Bragg diffraction. Energy and momentum conservation require: 39/44 Ω B = ω p ω s, k A = k p k s. Acoustic waves satisfy the dispersion relation: Ω B = v A k A 2v A k p sin(θ/2). In a single-mode fiber θ = 180, resulting in ν B = Ω B /2π = 2n p v A /λ p 11 GHz, if we use v A = 5.96 km/s, n p = 1.45, and λ p = 1.55 µm.

40 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 GeO 2 doping within the core. Multiple peaks are due to the excitation of different acoustic modes. Each acoustic mode propagates at a different velocity v A and thus leads to a different Brillouin shift (ν B = 2n p v A /λ p ).

41 Brillouin Threshold Pump and Stokes evolve along the fiber as di s dz = g BI p I s αi s, di p dz = g BI p I s αi p. Ignoring pump depletion, I p (z) = I 0 exp( αz). Solution of the Stokes equation: 41/44 I s (L) = I s (0)exp(g B I 0 L eff αl). Brillouin threshold is obtained from g B P th L eff A eff 21 = P th 21A eff g B L eff. Brillouin gain g B m/w is nearly independent of the pump wavelength.

42 Estimates of Brillouin 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 = µm 2. Threshold power P th 1 mw is relatively small. Yb-doped Fiber Lasers and Amplifiers Because of gain, L eff = [exp(gl) 1]/g > L. P th exceeds 20 W for a 1-m-long standard fibers. Further increase occurs for large-core fibers; P th 400 W when A eff 1000 µm 2. SBS is the dominant limiting factor at power levels P 0 > 1 kw. 42/44

43 Techniques for Controlling SBS Pump-Phase modulation: Sinusoidal modulation at several frequencies >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 ν B = 2n p v A /λ p through temperature dependence of n p. Built-in strain along the fiber: Changes in ν B 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 A eff. 43/44

44 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. 44/44 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.