314CHAPTER9.NOISEANDFREQUENCY CONTROL9.2Noise in Mode-locked LasersWithin this framework the response of the laser to noise can be easily in-cluded. The spontaneous emission noise due to the amplifying medium withsaturated gain gs and excess noise factor leads to additive white noise inthe perturbed master equation (9.6) with Lpert = (t,T), where is a whiteGaussian noisesource withautocorrelationfunction((t,T')s(t,T))=TPno(t-t)o(T-T)(9.18)where the spontaneous emission noise energy Pn -Tr withhweP, =02- we(9.19)-TRTpis added to the pulse within each roundtrip in the laser.T, is thecavity decaytime or photon lifetime in the cavity. Note, that the noise is approximatedby white noise. i.e. uncorrelated noise on both time scales t,T. The noisebetween different round-trips is certainly uncorrelated. However, white noiseon the fast time scale t, assumes a flat gain, which is an approximation.By projecting out the equations of motion for the pulse parameters in thepresence of this noise according to (9.8)-(9.13), we obtain the additionalnoise sources which are driving the energy, center frequency, tining andphase fluctuations in the mode-locked laser1aAw+Sw(T),(9.20)AwOTTwa20。Aw△A(T)(9.21)+ Se(T),OTRWoa1△p + Sp(T),(9.22)△p(T)OTTpa-2|DAt(9.23)Ap + St(T),OTTR
314 CHAPTER 9. NOISE AND FREQUENCY CONTROL 9.2 Noise in Mode-locked Lasers Within this framework the response of the laser to noise can be easily included. The spontaneous emission noise due to the amplifying medium with saturated gain gs and excess noise factor Θ leads to additive white noise in the perturbed master equation (9.6) with Lpert = ξ(t, T), where ξ is a white Gaussian noise source with autocorrelation function hξ(t 0 , T0 )ξ(t, T)i = T2 RPnδ(t − t 0 )δ(T − T0 ) (9.18) where the spontaneous emission noise energy Pn · TR with Pn = Θ2gs TR ~ωc = Θ~ωc τ p (9.19) is added to the pulse within each roundtrip in the laser. τ p is the cavity decay time or photon lifetime in the cavity. Note, that the noise is approximated by white noise, i.e. uncorrelated noise on both time scales t, T. The noise between different round-trips is certainly uncorrelated. However, white noise on the fast time scale t, assumes a flat gain, which is an approximation. By projecting out the equations of motion for the pulse parameters in the presence of this noise according to (9.8)—(9.13), we obtain the additional noise sources which are driving the energy, center frequency, timing and phase fluctuations in the mode-locked laser ∂ ∂T ∆w = − 1 τ w ∆w + Sw(T), (9.20) ∂ ∂T ∆θ(T) = 2φo TR ∆w wo + Sθ(T), (9.21) ∂ ∂T ∆p(T) = − 1 τ p ∆p + Sp(T), (9.22) ∂ ∂T ∆t = −2|D| TR ∆p + St(T), (9.23)
3159.2.NOISEINMODE-LOCKEDLASERSwith7-ReF(t)s(T,t)dtSu(T)(9.24)TR1-ReSe(T)f(t)s(T,t)dt(9.25)TR1Re(9.26)Sp(T)f,(t)s(T,t)dtTR1St(T)Reft(t)s(T,t)dt(9.27)TRThe new reduced noise sources obey the correlation functionsPas(T - T),(9.28)(Sw(T')Sw(T))4wo47元2）Pas(T - T),1 +(9.29)(Se(T")Se(T))3+12)wo4Pns(T - T),(9.30)<S,(T')S,(T)3WoPs(T - T),(9.31)<St(T')St(T))3wo(S;(T)S;(T))= 0fori+j.(9.32)The power spectra of amplitude, phase, frequency and timing fluctuationsare defined via the Fourier transforms of the autocorrelation functions[Aw(2)/2 =<Aw(T +T)Aw(T))e-jnTdt, etc.(9.33)After a short calculation, the power spectra due to amplifier noise are[A(2) /24Pn(9.34)1/+2. + 22 w。wo16dP72Pn14-3[40(2)2(1 +(9.35)2212Wo(1/+2 + 22) T w。14Pn(9.36)[Ap(2)-/21/t+023w。[At(2) 21[元? Pn4 4|D|2 Pn1(9.37)223wo(1/T2+22)3T-4W。<
9.2. NOISE IN MODE-LOCKED LASERS 315 with Sw(T) = 1 TR Re ½Z +∞ −∞ ¯f ∗ w(t)ξ(T,t)dt¾ , (9.24) Sθ(T) = 1 TR Re ½Z +∞ −∞ ¯f ∗ θ (t)ξ(T,t)dt¾ , (9.25) Sp(T) = 1 TR Re ½Z +∞ −∞ ¯f ∗ p (t)ξ(T,t)dt¾ , (9.26) St(T) = 1 TR Re ½Z +∞ −∞ ¯f ∗ t (t)ξ(T,t)dt¾ . (9.27) The new reduced noise sources obey the correlation functions hSw(T0 )Sw(T)i = Pn 4w0 δ(T − T0 ), (9.28) hSθ(T0 )Sθ(T)i = 4 3 µ 1 + π2 12¶ Pn wo δ(T − T0 ), (9.29) hSp(T0 )Sp(T)i = 4 3 Pn wo δ(T − T0 ), (9.30) hSt(T0 )St(T)i = π2 3 Pn wo δ(T − T0 ), (9.31) hSi(T0 )Sj (T)i = 0 for i 6= j. (9.32) The power spectra of amplitude, phase, frequency and timing fluctuations are defined via the Fourier transforms of the autocorrelation functions |∆wˆ(Ω)| 2 = Z +∞ −∞ h∆wˆ(T + τ)∆wˆ(T)ie−jΩτdτ, etc. (9.33) After a short calculation, the power spectra due to amplifier noise are ¯ ¯ ¯ ¯ ∆wˆ(Ω) wo ¯ ¯ ¯ ¯ 2 = 4 1/τ 2 w + Ω2 Pn wo , (9.34) |∆ˆθ(Ω)| 2 = 1 Ω2 ∙ 4 3 µ 1 + π2 12¶ Pn wo + 16 (1/τ 2 p + Ω2) φ2 o T2 R Pn wo ¸ , (9.35) |∆pˆ(Ω)τ | 2 = 1 1/τ 2 p + Ω2 4 3 Pn wo , (9.36) ¯ ¯ ¯ ¯ ∆t ˆ(Ω) τ ¯ ¯ ¯ ¯ 2 = 1 Ω2 ∙ π2 3 Pn wo + 1 (1/τ 2 ω + Ω2) 4 3 4|D| 2 T2 Rτ 4 Pn wo ¸ . (9.37)
316CHAPTER9.NOISEANDFREQUENCYCONTROLThese equations indicate, that energy and center frequency fluctuations be-come stationary with mean square fuctuationsPnTw(9.38)Wo2PhT2(AWT)2) :(9.39)3wowhereas the phase and timing undergo a random walk with variances(1+) P/T)+ge(T) = 《(A(T) - △(0)2) =(9.40)12313Φ Pn[T]T+16ATwTpTw△t(T) -△t(0)2 PnTl(9.41)ot(T)3woT4 4|D|2 Pn[T]][T]3T-4WoTpTpThe phasenoise causes thefundamental finite width of every lineof themode-locked comb in the optical domain. The timing jitter leads to a fi-nitelinewidth of thedetectedmicrowavesignal.whichisequivalenttothelasers fundamental fluctuations in repetition rate.In the strict sense, phaseand timing in a free running mode-locked laser (or autonomous oscillator)are not anymore stationary processes. Nevertheless, since we know theseare Gaussian distributed variables.wecan computethe amplitude spectra ofphasors undergoing phase diffusion processes rather easily. The phase differ-ence =△e(T) -△e(O) is a Gaussian distributed variable with variance andpropabilitydistribution, withg=(2)(9.42)p() :V2元0Therefore, the expectation value of a phasor with phase is(ejp)ejdp(9.43)V2元g=e-10
316 CHAPTER 9. NOISE AND FREQUENCY CONTROL These equations indicate, that energy and center frequency fluctuations become stationary with mean square fluctuations *µ∆w wo ¶2 + = 2 Pnτ w wo (9.38) h(∆ωτ ) 2 i = 2 3 Pnτ 2 p wo (9.39) whereas the phase and timing undergo a random walk with variances σθ(T) = h(∆θ(T) − ∆θ(0))2 i = 4 3 µ 1 + π2 12¶ Pn wo |T| (9.40) +16 φ2 o T2 R Pn wo τ 3 p µ exp ∙ −|T| τ p ¸ − 1 + |T| τ w ¶ σt(T) = *µ∆t(T) − ∆t(0) τ ¶2 + = π2 3 Pn wo |T| (9.41) + 4 3 4|D| 2 T2 Rτ 4 Pn wo τ 3 ω µ exp ∙ −|T| τ p ¸ − 1 + |T| τ p ¶ The phase noise causes the fundamental finite width of every line of the mode-locked comb in the optical domain. The timing jitter leads to a fi- nite linewidth of the detected microwave signal, which is equivalent to the lasers fundamental fluctuations in repetition rate. In the strict sense, phase and timing in a free running mode-locked laser (or autonomous oscillator) are not anymore stationary processes. Nevertheless, since we know these are Gaussian distributed variables, we can compute the amplitude spectra of phasors undergoing phase diffusion processes rather easily. The phase difference ϕ = ∆θ(T) − ∆θ(0) is a Gaussian distributed variable with variance σ and propability distribution p(ϕ) = 1 √2πσ e− ϕ2 2σ , with σ = ϕ2® . (9.42) Therefore, the expectation value of a phasor with phase ϕ is ejϕ® = 1 √2πσ Z +∞ −∞ e− ϕ2 2σ ejϕdϕ (9.43) = e− 1 2 σ