Self Phase Modulation (SPM)

Self Phase Modulation (SPM) Nonlinear interaction of the optical field with itself in the fiber Starting from the Kerr effect n ' = Re( ε ) = n + nNL E = n + nNL nNL yields a nonlinear phase constant ∆β k0 ∫ ∫ ∆ n F ( x , y ) dxdy −∞ −∞ +∞ +∞ ( β (ω ) = β (ω ) + ∆β ) +∞ +∞ ∆β = P Aeff ∫ ∫ F ( x , y ) dxdy ∆ n = nNL P Aeff −∞ −∞ Given a time varying optical field, E(t) and power P(t), nNL is responsible for a nonlinear phase φNL For a pulse, this nonlinear phase varies on the time scale of the pulse width A time varying phase results in a change in carrier frequency namely the optical frequency ω0 changes by δω, also on the time scale of the pulse ∆ω and φΝL are related in the usual manner ∂φ NL δω (t ) = − ∂t The minus sign is chosen so as to be consistent with the notation exp(-iω0t) SPM induced spectral broadening The time dependence of φNL is responsible for spectral broadening The temporal phase change implies that the instantaneous optical frequency differs across the pulse from its central value ω0 The frequency change is determined by the rate of change of the power δω (t ) = − Linear regime Unchirped pulse ∂φ NL ∂t Non linear regime Chirped pulse ∂P > φ NL > δω < ∂t Leading edge Red shift ∂P < φ NL < δω > ∂t Trailing edge Blue shift Spectral modification due to SPM Non linear regime Chirped pulse Linear regime Unchirped pulse ∂P > φ NL > δω < ∂t Leading edge Red shift ∂P < φ NL < δω > ∂t Trailing edge Blue shift Nonlinear Schrödinger Equation : ∂A + i β ∂ A + α A = iγ A A 2 ∂z ∂t In the general case, the nonlinear Schrödinger equation has to be solved In the presence of dispersion the solution may be cumbersome and often needs to be numerical In the operation regimes which have been defined, in the regime where the dispersion plays but a minor role, β2 is set to zero and then for: A( z,τ ) = P0 exp( −α z / 2)U ( z,τ ) ∂U e −αz =i U U ∂z LNL U = Ve iφ NL yields LNL = τ= t T0 γP0 ∂V ∂ φ NL exp( −α z ) = and = V ∂z ∂z L NL U ( L, t ) = U (0, t ) exp(iφ NL ( L, t )) φ NL ( L, t ) = U (0, t ) Effective fiber length L Leff = ∫ e −α z dz = 1− e −α z α ≈ Leff LNL Lef z z > 1/α f 1/α z z The maximum nonlinear phase shift occurs at the maximum power point namely at the pulse peak φmax = Leff / LNL = γP0 Leff The nonlinear length is the propagation distance for which the accumulated maximum nonlinear phase φmax =1 The frequency change due to a time varying nonlinear phase is δω (t ) = − ⎛L ⎞ ∂ ⎛ Leff ⎞ ∂ ∂φ NL ⎟⎟ U (0, t ) = −⎜⎜ eff ⎟⎟ ⎛⎜ U (0, t ) ⎞⎟ = −⎜⎜ ⎠ ∂t ⎝ LNL ⎠ ∂t ⎝ ⎝ LNL ⎠ ∂t SPM induced pulse broadening: normal dispersion regime (D < 0) Dispertion (ps/(km-nm)) 30 Leading edge δω < 25 Trailing edge δω > λ d = 31 µ m 20 15 10 -5 -10 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 Wavelength Leading edge δω < δλ > D δλ < D → δβ1 > β1 increases → V g decreases The leading edge travels fast The trailing edge travels slow The pulse broadens since the leading edge propagates at an increased Vg while the trailing edge propagates at a decreasing Vg This broadening adds to the linear dispersive broadening The nonlinear frequency shift depends on the rate of phase change which is determined by the rate of power change A steep pulse edge causes larger nonlinear frequency shifts Super Gaussian pulses ⎛ ⎛ t ⎞2 N ⎞ U (0, t ) = exp⎜ − ⎜⎜ ⎟⎟ ⎟ ⎜ T0 ⎟ ⎝ ⎠ ⎠ ⎝ Normalized power N=1 N=2 N=3 N=4 -3 -2 -1 t / T0 The SPM induces frequency chirp for a super Gaussian pulse is : Phase φNL N Leff ⎛ t ⎞ ⎜⎜ ⎟⎟ δω ( t ) = T0 L NL ⎝ T0 ⎠ N −1 ⎡ ⎛ t exp ⎢ − ⎜⎜ ⎢ ⎝ T0 ⎣ ⎞ ⎟⎟ ⎠ 2N ⎤ ⎥ ⎥ ⎦ N=3 N=1 0.8 0.6 0.4 0.2 δωT0 Frequency chirp -2.5 -1 -2 -3 -2.5 -2 -1.5 -1 -0.5 t/T0 -1.5 -1 1.5 2.5 1.5 2.5 N=3 N=1 -2 0.5 -0.5 t/T0 0.5 Characteristics of SPM induced chirp • δω < at the pulse leading edge (red shift) • δω > at the pulse trailing edge (blue shift) • The chirp is larger for pulses with steeper transitions • For Gaussian pulses, the chip is linear and positive over a large central region of the pulse • For super Gaussian pulse, the chirp only occurs at the transitions and is not linear anywhere • For D < (normal dispersion) SPM together with dispersion lead to pulse broadening The optical pulse power spectrum is given by S ( L, ω ) = A( L, ω ) ∞ S ( L, ω ) ∫ A(0, t ) exp[iφ NL ( L, t ) + iωt ]dt −∞ The temporal phase changes cause new frequency components which are generated continuously namely spectral broadening takes place If there were no dispersion (D=0) the temporal pulse shape would remain unchanged However, a finite dispersion causes pulse broadening (for D < 0) and to soliton effects for D > The pulse spectrum comprises a multi peak structure due to SPM The number of peaks M in the SPM- broadened spectrum is approximately given by : φmax~(M-0.5) π For super Gaussian pulses, down shifted frequencies are generated at the up-going transition while up shifted frequencies are generated at the down-going transition No frequency chirp is generated inside the pulse (where the power is constant) Gaussian pulse field in time domain 0.5 Phimax=3π Phimax=3π Phimax=0 Phimax=6π -0.5 -1 -50 -40 -30 -20 -10 -1 10 20 30 40 50-50 -40 -30 -20 -10 -1 10 20 30 40 50 -50 -40 -30 -20 -10 time in ps time in ps time in ps 10 20 30 40 50 Gaussian pulse envelope spectrum Phimax=0 Phimax=6π S(z,w)/S(0,w) Phi =3π Phi =3π max max -400 -200 200 400 optical frequency detuning in GHz -400 -200 200 optical frequency detuning in GHz 400 -400 -200 200 optical frequency detuning in GHz 400 Super Gaussian (N=2) pulse field in time domain Phi Phimax =3π =3π max Phimax=6π 0.5 Phimax=0 -0.5 -1 -40 -30 -20 -10 10 20 30 40 -40 -30 -20 -10 time in ps 10 20 30 40 -40 -30 -20 -10 time in ps 10 20 30 40 time in ps Super Gaussian (N=2) pulse envelope spectrum 0.4 Phimax=6π S(z,w)/S(0,w) 0.6 Phimax=3π Phimax=0 0.8 0.2 -400 -200 200 400 -400 optical frequency detuning in GHz -200 200 optical frequency detuning in GHz 400 -400 -200 200 optical frequency detuning in GHz 400