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Nonlinear Effects in Optical Fibers

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Nonlinear Effects in Optical Fibers
Govind P. Agrawal
Institute of Optics
University of Rochester
Rochester, NY 14627




c 2006 G. P. Agrawal                 Back
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Outline
• Introduction                      2/100


• Stimulated Raman Scattering
• Stimulated Brillouin Scattering
• Self-Phase Modulation
• Cross-Phase Modulation
• Four-Wave Mixing
• Supercontinuum Generation
• Concluding Remarks                Back
                                    Close
Introduction
Fiber nonlinearities                               3/100


 • 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.
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Major Nonlinear Effects
  • Stimulated Raman Scattering (SRS)             4/100

  • 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.
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Stimulated Raman Scattering
 • Scattering of light from vibrating silica molecules.
                                                                     5/100
 • 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.
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SRS Dynamics
 • SRS process is governed by two coupled equations:
                                                                         6/100
             dIp                         dIs
                 = −gRIpIs − α pIp,          = gRIpIs − αsIs.
             dz                          dz
 • If we neglect pump depletion (Is Ip), pump power decays
   exponentially, and the Stokes beam satisfies
                        dIs
                            = gRI0e−α pzIs − αsIs.
                        dz
 • This equation has the solution

   Is(L) = Is(0) exp(gRI0Leff − αsL),     Leff = [1 − exp(−α pL)]/α 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
                                                                           7/100
   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:
                      ∞
           Ps(L) =        hω exp[gR(ω p − ω)I0Leff − αsL] dω.
                          ¯
                     −∞

 • Approximate solution (using the method of steepest descent):
                          eff
                 Ps(L) = Ps0 exp[gR(ΩR)I0Leff − αsL].
 • Effective input power is given by
                                                 1/2            −1/2
           eff                           2π            ∂ 2 gR
          Ps0 = hωsBeff,
                ¯              Beff =                                  .
                                        I0Leff         ∂ ω2     ω=ωs       Back
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Raman Threshold
 • Raman threshold is defined as the input pump power at which
                                                                      8/100
   Stokes power becomes equal to the pump power at the fiber output:

                   Ps(L) = Pp(L) ≡ P0 exp(−α pL).

 • P0 = I0Aeff is the input pump power.
 • For αs ≈ α p, threshold condition becomes
                      eff
                     Ps0 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                       16Aeff
                            ≈ 16   →      Pth ≈          .
                    Aeff                          gRLeff
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Estimates of Raman Threshold
Telecommunication Fibers
                                                                   9/100
  • 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
  • For short fibers (L < 100 m), Leff = L.
  • For fibers with a large core, Aeff ∼ 500 µm2.
  • Pth can exceed 100 kW depending on fiber length.
  • SRS may limit fiber lasers and amplifiers if L   10 m.           Back
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SRS: Good or Bad?
 • Raman gain introduces interchannel crosstalk in WDM systems.
                                                                       10/100
 • 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.

                                                                       Back
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Raman Amplifiers
 • Pumped in backward direction using diode lasers.
                                                                         11/100
 • 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.
                                                                         12/100
 • 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.   Back
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Brillouin Shift
 • Pump produces density variations through electrostriction, resulting
                                                                          13/100
   in an index grating which generates Stokes wave through Bragg
   diffraction.
 • 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.
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Brillouin Gain Spectrum
 • Decay of acoustic waves as exp(−ΓBt) leads to a Lorentzian gain
                                                                      14/100
   spectrum of the form
                                     (ΓB/2)2
                   gB(Ω) = g p                     .
                               (Ω − ΩB)2 + (ΓB/2)2
 • Peak gain depends on the material parameters as
                                       8π 2γe2
                    g p ≡ gB(ΩB) =      2
                                                  .
                                   n pλ p ρ0cvAΓB

 • Electrostrictive constant γe = ρ0(dε/dρ)ρ=ρ0 ≈ 0.902 for silica.
                                          −2
 • Gain bandwidth ΓB scales with λ p 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
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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 modes.
 • Each acoustic mode propagates at a different velocity vA and thus
   leads to a different Brillouin shift (νB = 2n pvA/λ p).
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Brillouin Threshold
 • Pump and Stokes evolve along the fiber as
                                                                    16/100
                dIs                       dIp
            −       = gBIpIs − αIs,           = −gBIpIs − αIp.
                dz                        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                        21Aeff
                             ≈ 21 →         Pth ≈          .
                      Aeff                          gBLeff

 • Brillouin gain gB ≈ 5 × 10−11 m/W is nearly independent of the
   pump wavelength.                                                 Back
                                                                    Close
Estimates of Brillouin Threshold
Telecommunication Fibers
                                                                     17/100
  • 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
  • For short fibers (L < 100 m), Leff = L.
  • Pth exceeds 20 W for a 1-m-long fiber.
  • Further increase occurs for large-core fibers; Pth ∼ 200 W when
    Aeff ∼ 500 µm2.
  • SBS is the dominant limiting factor at power levels P0 > 1 kW.   Back
                                                                     Close
Techniques for Controlling SBS
 • Pump-Phase modulation: Sinusoidal modulation at several frequen-
                                                                      18/100
   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 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: Wider core reduces SBS threshold by en-
   hancing Aeff.                                                      Back
                                                                      Close
Fiber Gratings for Controlling SBS
 • Fiber Bragg gratings can be employed for SBS suppression [Lee and
                                                                         19/100
   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.     Back
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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.
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Self-Phase Modulation
 • Refractive index 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.
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Nonlinear Phase Shift
 • Pulse propagation governed by Nonlinear Schr¨dinger Equation
                                               o
                                                                  22/100
                        ∂ A β2 ∂ 2A
                      i    −      2
                                    + γ|A|2A = 0.
                        ∂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.

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SPM-Induced Chirp
                                                       23/100




   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
                                                                        24/100
   lightwave system.
 • Modulation instability often enhances system noise.

   On the positive side . . .
 • Modulation instability can be used to produce ultrashort pulses at
   high 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
               o
 Nonlinear Schr¨dinger Equation
                                                             25/100
                      ∂ A β2 ∂ 2A
                    i    −      2
                                  + γ|A|2A = 0.
                      ∂z   2 ∂t




 • CW solution unstable for anomalous dispersion (β2 < 0).
 • Useful for producing ultrashort pulse trains.
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Modulation Instability
 • A CW beam can be converted into a pulse train.
                                                                     26/100
 • 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
                                                              27/100




 • An example of the Sagnac interferometer.
 • Transmission through the fiber loop:

                 T = 1 − 4 f (1 − f ) cos2[( f − 1 )γP0L].
                                                 2

 • f = fraction of power in the CCW direction.
 • T = 0 for a 3-dB coupler (loop acts as a perfect mirror)
 • Power-dependent transmission for f = 0.5.
                                                              Back
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Passive Mode Locking
                                                            28/100




 • Figure-8 fiber laser can produce pulses ∼100 fs.
 • Amplifier located asymmetrically inside the NFLM.
 • SPM-induced phase shift larger in clockwise direction.
 • Low-power light reflected by the loop.
 • Central part of the pulse transmitted.
 • Transmitted pulses become narrower.
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Cross-Phase Modulation
 • Consider two optical fields propagating simultaneously.
                                                                       29/100
 • 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:

                   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.
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XPM-Induced Chirp
 • Fiber dispersion affects the XPM considerably.
                                                           30/100
 • 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.
                                                          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 switching
  • Demultiplexing of OTDM channels
  • Wavelength conversion of WDM channels
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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 pulse.
 • Can produce ∼100 fs pulses.
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Synchronous Mode Locking
                                                                     33/100




 • 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
                                                                           34/100




 • A Mach–Zehnder 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 (<1 ps).
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Four-Wave Mixing (FWM)
                                                                      35/100




 • FWM is a nonlinear process that transfers energy of pumps
   to signal and idler waves.
 • FWM requires conservation of (notation: E = Re[A exp(iβ z−iωt)])
      Energy              ω1 + ω2 = ω3 + ω4
      Momentum           β1 + β2 = β3 + β4
 • Degenerate FWM:     Single pump (ω1 = ω2).
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FWM: Good or Bad?
 • FWM leads to interchannel crosstalk in WDM systems.
                                                                    36/100
 • 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.
                                                                     37/100
 • Pump power is transferred to signal through FWM.
 • Peak gain G p = 1 exp(2γP0L) can exceed 20 dB for
                   4
   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.


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   Four-Wave Mixing FOPAs
Single- and Dual-PumpFour-Wave
                      (FWM)                                  Mixing (FW
                                                                     38/100
              Pump


                                           Pump 1         Pump 2

     Signal               Idler
                                            Signal        Idler




      λ3      λ1          λ4             λ1 λ3       λ0   λ4 λ2


  • Pump close       to      fiber’s   • Pumps at opposite ends
    ZDWL                              • Much more uniform gain
  • Wide but nonuniform gain          • Lower pump powers (∼0.5 W)
    spectrum with a dip
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Optical Phase Conjugation
 • FWM generates an idler wave during parametric amplification.
                                                                         39/100
 • Its phase is complex conjugate of the signal field (A4 ∝ A∗) because
                                                            3
   of spectral 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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Wavelength Conversion
                                                                       40/100




 • FWM can transfer data to a different wavelength.
 • A CW pump beam is launched into the fiber together with the signal
   channel.
 • Its wavelength is chosen half way from the desired shift.
 • FWM transfers the data from signal to the idler beam at the new
   wavelength.
                                                                       Back
                                                                       Close
Highly Nonlinear Fibers
 • Silica nonlinearity is relatively weak (n2 = 2.6 × 10−20 m2/W).
                                                                        41/100
 • Applications of nonlinear effects require high input powers in com-
   bination with long fiber lengths (> 1 km).
 • Parameter γ = 2πn2/(λ Aeff) can be increased by reducing Aeff.
 • Such fibers are called highly nonlinear fibers. Examples include
   photonic-crystal, tapered, and other microstructure fibers.




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Supercontinuum Generation
  • FWM in combination with SPM, XPM, and SRS can generate su-
                                                                  42/100
    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-
                                                                           43/100
   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.
                                                                         44/100
 • 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.
                                                                         Back
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