Chapter 9Noise and Frequency ControlSo far we only considered the deterministic steady state pulse formation inultrashort pulse laser systems due to the most important pulse shaping mech-anisms prevailing in todays femtosecond lasers. Due to the recent interestin using modelocked lasers for frequency metrology and high-resolution laserspectroscopy as well as phase sensitive nonlinear optics the noise and tuningproperties of mode combs emitted by modelocked lasers is of much currentinterest. Soliton-perturbation theory is well suited to successfully predictthe noise behavior of many solid-state and fiber laser systems [1] as well aschanges in group- and phase velocity in modelocked lasers due to intracavitynonlinear effects [5]. We start off by reconsidering the derivation of the mas-ter equation for describing the pulse shaping effects in a mode-locked laser.We assume that in steady-state the laser generates at some position z (forexample at the point of the output coupler) inside the laser a sequence ofpulses with the envelope a(T = mTr,t). These envelopes are the solutionsof thecorresponding masterequation,where thedynamics per roundtrip isdescribed on a slow time scale T = mTr. Then the pulse train emitted fromthe laser including the carrier isA(T,t) = Z a(T = mT,t)e[e(t-mTR+(言-)2mL).(9.1)m=-αwith repetition rate fr = 1/Tr and center frequency we. Both are in generalsubject to slow drifts due to mirror vibrations, changes in intracavity pulseenergy that might befurther converted into phase and group velocity changes.Note, the center frequency and repetition rate are only defined for times long309
Chapter 9 Noise and Frequency Control So far we only considered the deterministic steady state pulse formation in ultrashort pulse laser systems due to the most important pulse shaping mechanisms prevailing in todays femtosecond lasers. Due to the recent interest in using modelocked lasers for frequency metrology and high-resolution laser spectroscopy as well as phase sensitive nonlinear optics the noise and tuning properties of mode combs emitted by modelocked lasers is of much current interest. Soliton-perturbation theory is well suited to successfully predict the noise behavior of many solid-state and fiber laser systems [1] as well as changes in group- and phase velocity in modelocked lasers due to intracavity nonlinear effects [5]. We start off by reconsidering the derivation of the master equation for describing the pulse shaping effects in a mode-locked laser. We assume that in steady-state the laser generates at some position z (for example at the point of the output coupler) inside the laser a sequence of pulses with the envelope a(T = mTr, t). These envelopes are the solutions of the corresponding master equation, where the dynamics per roundtrip is described on a slow time scale T = mTR. Then the pulse train emitted from the laser including the carrier is A(T,t) = X +∞ m=−∞ a(T = mTr, t)e j h ωc ³ (t−mTR+ ³ 1 vg − 1 vp ´ 2mL´i . (9.1) with repetition rate fR = 1/TR and center frequency ωc. Both are in general subject to slow drifts due to mirror vibrations, changes in intracavity pulse energy that might be further converted into phase and group velocity changes. Note, the center frequency and repetition rate are only defined for times long 309
310CHAPTER9.NOISEANDFREQUENCYCONTROLcompared to the roundtrip time in the laser. Usually, they only change ona time scale three orders of magnitude longer than the expectation value ofthe repetition rate.9.1The Mode CombLets suppose the pulse envelope, repetition rate, and center frequency do notchange any more. Then the corresponding time domain signal is sketched inFigure 9.1.VgIA(t)/p1fCEFigure 9.1: Pulse train emitted from a noise free mode-locked laser. Thepulses can have chirp. The intensity envelope repeats itself with repetitionrate fr.The electric field is only periodic with the rate fce if it is related tothe repetion rate by a rational number.The pulse a(T = mTr,t) is the steady state solution of the master equa-tion describing the laser system, as studied in chapter 6. Let's assume thatthe steady state solution is a purturbed soliton according to equation (6.64).(A ech(*二t) + ae(T,t)-j0oTRa(t,T) =(9.2)with the soliton phase shiftID](9.3)OA00-2
310 CHAPTER 9. NOISE AND FREQUENCY CONTROL compared to the roundtrip time in the laser. Usually, they only change on a time scale three orders of magnitude longer than the expectation value of the repetition rate. 9.1 The Mode Comb Lets suppose the pulse envelope, repetition rate, and center frequency do not change any more. Then the corresponding time domain signal is sketched in Figure 9.1. Figure 9.1: Pulse train emitted from a noise free mode-locked laser. The pulses can have chirp. The intensity envelope repeats itself with repetition rate fR. The electric field is only periodic with the rate fCE if it is related to the repetion rate by a rational number. The pulse a(T = mTr, t) is the steady state solution of the master equation describing the laser system, as studied in chapter 6. Let’s assume that the steady state solution is a purturbed soliton according to equation (6.64). a(t, T) = µ A0 sech( t − t0 τ ) + ac(T,t) ¶ e −jφ0 T TR (9.2) with the soliton phase shift φ0 = 1 2 δA2 0 = |D| τ 2 (9.3)
3119.1.THEMODE COMBThus, there is a carrier envelope phase shift Aoce from pulse to pulse givenby12L - Φo +mod(2元)△OcE(9.4)gUgWTRΦo+mod(2元)UrTheFourier transform of theunperturbed pulse train isei(AOcE-(w-we)TR)mA(w) = a(w-we)>m== a(w-we) ejmTn(-u)FOPCE>= a(w-we)TRO(9.5)TRYwhich is shown in Figure 9.2.Each comb line is shifted by the carrier-envelopeoffsetfrequencyfcE=fromtheorigin2元1BA(f-fo)0f,=foe+nf.Figure 9.2: Opitcal mode comb of a mode-locked laser output.To obtain self-consistent equations for the repetition rate, center fre-quency and the otherpulse parameters we employ soliton-perturbation the-Ory. This is justified for the case, where the steady state pulse is close to a
9.1. THE MODE COMB 311 Thus, there is a carrier envelope phase shift ∆φCE from pulse to pulse given by ∆φCE = µ 1 vg − 1 vp ¶¯ ¯ ¯ ¯ ωc 2L − φ0 + mod(2π) (9.4) = ωcTR µ 1 − vg vp ¶ − φ0 + mod(2π) The Fourier transform of the unperturbed pulse train is Aˆ(ω)=ˆa(ω − ωc) X +∞ m=−∞ ej(∆φCE−(ω−ωc)TR)m = ˆa(ω − ωc) X +∞ n=−∞ e jmTR ³ ∆φCE TR −ω ´ = ˆa(ω − ωc) X +∞ n=−∞ TRδ µ ω − µ∆φCE TR + nωR ¶¶ (9.5) which is shown in Figure 9.2. Each comb line is shifted by the carrier-envelope offset frequency fCE = ∆φCE 2πTR from the origin Figure 9.2: Opitcal mode comb of a mode-locked laser output. To obtain self-consistent equations for the repetition rate, center frequency and the other pulse parameters we employ soliton-perturbation theory. This is justified for the case, where the steady state pulse is close to a
312CHAPTER9.NOISEANDFREQUENCYCONTROLsoliton,i.e.for thefast saturable absorber case, this is the chirp free solution,occuring when the ratio of gain filtering to dispersion is equal to the ratioof SAM action to self-phase modulation, see Eq. (6.61). Then the pulsesolution in the m-th roundtrip is a solution of the nonlinear SchrodingerEquation stabilized by the irreversible dynamics and subject to additionalperturbations2aA-APATR2'aOt2(9.6)02+(g-1)A+Di%2A+/APA+LpertDue to the irreversible processes and the perturbations the solution to (9.6)is a soliton like pulse with perturbations in amplitude,phase, frequency andtiming plus some continuumA(t,T) = [(Ao + △A。) sech[(t-△t)/] +ac(T,t)](9.7)e-jT/TRejAp(T)te-j00with pulse energy wo = 2A?t.The perturbations cause fuctuations in amplitude, phase, center fre-quency and timing of the soliton and generate background radiation, i.e.continuum△A(T,t) = △w(T)fw(t) +△e(T)fe(t) +p(T)fp(t)(9.8)+△t(T)fi(t) + ae(T,t).where, we rewrote the amplitude perturbation as an energy perturbationThedynamics of thepulseparameters due to the perturbed Nonlinear SchrodingerEquation (9.6) can be projected out from the perturbation using the adjointbasis and the orthogonality relation, see Chapter 3.5. Note, that the f; cor-respond to the first component of the vector in Eqs.(3.22) - (3.25). The dy-namics of the pulse parameters due to the perturbed Nonlinear SchrodingerEquation (9.6) can be projected out from the perturbation using the adjointbasis f* corresponding to the first component of the vector in Eqs.(3.44) -(3.47) and the new orthogonality relationRef(t)fi(t)dt /=i .(9.9)
312 CHAPTER 9. NOISE AND FREQUENCY CONTROL soliton, i.e. for the fast saturable absorber case, this is the chirp free solution, occuring when the ratio of gain filtering to dispersion is equal to the ratio of SAM action to self-phase modulation, see Eq. (6.61). Then the pulse solution in the m−th roundtrip is a solution of the nonlinear Schrödinger Equation stabilized by the irreversible dynamics and subject to additional perturbations TR ∂ ∂T A = jD ∂2 ∂t2A − jδ|A| 2 A +(g − l)A + Df ∂2 ∂t2A + γ|A| 2 A + Lpert (9.6) Due to the irreversible processes and the perturbations the solution to (9.6) is a soliton like pulse with perturbations in amplitude, phase, frequency and timing plus some continuum A(t, T) = [(Ao + ∆Ao ) sech[(t − ∆t)/τ ] + ac(T,t)] e−jφoT /TR ej∆p(T)t e−jθ0 (9.7) with pulse energy w0 = 2A2 oτ . The perturbations cause fluctuations in amplitude, phase, center frequency and timing of the soliton and generate background radiation, i.e. continuum ∆A(T,t) = ∆w(T)fw(t) + ∆θ(T)fθ(t) + ∆p(T)fp(t) +∆t(T)ft(t) + ac(T,t). (9.8) where, we rewrote the amplitude perturbation as an energy perturbation. The dynamics of the pulse parameters due to the perturbed Nonlinear Schrödinger Equation (9.6) can be projected out from the perturbation using the adjoint basis and the orthogonality relation, see Chapter 3.5. Note, that the fi correspond to the first component of the vector in Eqs.(3.22) - (3.25). The dynamics of the pulse parameters due to the perturbed Nonlinear Schrödinger Equation (9.6) can be projected out from the perturbation using the adjoint basis ¯f ∗ i corresponding to the first component of the vector in Eqs.(3.44) - (3.47) and the new orthogonality relation Re ½Z +∞ −∞ ¯f ∗ i (t)fj (t)dt¾ = δi,j . (9.9)
3139.1.THEMODE COMBWe obtaina11AwAu-ReF(t)Lpert(T,t)dt(9.10)OTTRTwXa120。w△(T)-ReF(t)Lpert(T,t)dt(9.11)OTTRWoTR011Fs(t)Lpert(T,t)dt !ApRe△p(T)(9.12)OTTRTpa-2DI1Atf:(t)Lpert(T,t)dt )NRe(9.13)OTTRTRNote, that the irreversible dynamics does couple back the generated con-tinuum to the soliton parameters. Here, we assume that this coupling issmall and neglect it in the following, see [1]. Due to gain saturation and theparabolic filter pulse energy and center frequency fuctuations are dampedwithnormalizeddecayconstants1= (2gd- 2A)(9.14)Tw1149s(9.15)Tp-302T2TRHere, gs is the saturated gain and gd is related to the differential gain by9o(9.16)gs =1+ Pdgs(9.17).Wogd=dwoNote, in this model we assumed that the gain instantaneously follows theintracavity average power or pulse energy, which is not true in general. How-ever, it is straight forward to include the relaxation of the gain by adding adynamical gain model to the perturbation equations. For simplicity we shallneglect this here. Since the system is autonomous, there is no retiming andrephasing in the free running system
9.1. THE MODE COMB 313 We obtain ∂ ∂T ∆w = − 1 τ w ∆w + 1 TR Re ½Z +∞ −∞ ¯f ∗ w(t)Lpert(T, t)dt¾ (9.10) ∂ ∂T ∆θ(T) = 2φo TR ∆w wo + 1 TR Re ½Z +∞ −∞ ¯f ∗ θ (t)Lpert(T,t)dt¾ (9.11) ∂ ∂T ∆p(T) = − 1 τ p ∆p + 1 TR Re ½Z +∞ −∞ ¯f ∗ p (t)Lpert(T,t)dt¾ (9.12) ∂ ∂T ∆t = −2|D| TR ∆ω + 1 TR Re ½Z +∞ −∞ ¯f ∗ t (t)Lpert(T,t)dt¾ (9.13) Note, that the irreversible dynamics does couple back the generated continuum to the soliton parameters. Here, we assume that this coupling is small and neglect it in the following, see [1]. Due to gain saturation and the parabolic filter pulse energy and center frequency fluctuations are damped with normalized decay constants 1 τ w = (2gd − 2γA2 o) (9.14) 1 τ p = 4 3 gs Ω2 gτ 2 1 TR (9.15) Here, gs is the saturated gain and gd is related to the differential gain by gs = go 1 + wo PLTR (9.16) gd = dgs dwo · wo (9.17) Note, in this model we assumed that the gain instantaneously follows the intracavity average power or pulse energy, which is not true in general. However, it is straight forward to include the relaxation of the gain by adding a dynamical gain model to the perturbation equations. For simplicity we shall neglect this here. Since the system is autonomous, there is no retiming and rephasing in the free running system
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
3179.2.NOISEINMODE-LOCKED LASERS9.2.1 The Optical Spectruma(t, T) = (Ao sech(*=fo) + a(T,t)e-jgoT(9.44)In the presence of noise the laser output changes from eq.(9.1) to a randomprocess.Neglecting the background continuum we obtain:t-mTR-△t(mTR)A(t,T = mTr)= Z (Ao+AA(mTr)sechTejApcE-me(we+Ap(mTn)te-jAo(mTr)For simplicity, we will neglect in the following amplitude and carrier fre-quency fluctuations in Eq.(9.45), because they are bounded and become onlyimportant at large offsets from the comb. However, we keep them in the ex-pressions for the phase and timing jitter Eqs.(9.34) and (9.36). We assumea stationary process, so that the optical power spectrum can be computedfromlim(At(w)Ar(w)(9.46)S(w) = T=2NTRwith the spectra related to a finite time intervalAr(u) = A(t)e-jiutdt = ao(w -we) m=-nejmTr(-(9.47)e-i[(w-we)At(mTR)+A(mTr)where ao(w) is the Fourier transform of the pulse shape. In this case Ao sech) e-jut dt = AoTT sech ((9.48)ao(w) =:12With (9.46) the optical spectrum of the laser is given byF-w）(m-mejTR("FRS() limlo(we)-M--(e+j[2(f-fe)(△t(mTr)-t(m*Tr)-(mTr)-(m'Tr)(9.49)
9.2. NOISE IN MODE-LOCKED LASERS 317 9.2.1 The Optical Spectrum a(t, T) = µ A0 sech( t − t0 τ ) + ac(T, t) ¶ e −jφ0 T TR (9.44) In the presence of noise the laser output changes from eq.(9.1) to a random process. Neglecting the background continuum we obtain: A(t, T = mTR) = X +∞ m=−∞ (A0 + ∆A(mTR)) sec h µt − mTR − ∆t(mTR) τ ¶ (9.45) ej∆φCE·mej(ωc+∆p(mTR))t e−j∆θ(mTR) For simplicity, we will neglect in the following amplitude and carrier frequency fluctuations in Eq.(9.45), because they are bounded and become only important at large offsets from the comb. However, we keep them in the expressions for the phase and timing jitter Eqs.(9.34) and (9.36). We assume a stationary process, so that the optical power spectrum can be computed from S(ω) = lim T =2NTR→∞ 1 T hAˆ∗ T (ω)AˆT (ω)i (9.46) with the spectra related to a finite time interval AˆT (ω) = R T −T A(t)e−jωtdt = ˆa0(ω − ωc) PN m=−N e jmTR ³ ∆φCE TR −ω ´ e−j[(ω−ωc)∆t(mTR)+∆θ(mTR)] (9.47) where aˆ0(ω) is the Fourier transform of the pulse shape. In this case aˆ0(ω) = Z ∞ −∞ A0 sec h µ t τ ¶ e−jωtdt = A0πτ sec h ³π 2 ωτ´ (9.48) With (9.46) the optical spectrum of the laser is given by S(ω) = limN→∞ |aˆ0(ω − ωc)| 2 1 2NT R PN m0=−N PM m=−N e jTR ³ φCE TR −ω ´ (m−m0 ) he+j[2π(f−fc)(∆t(mTR)−∆t(m0 TR))−(θ(mTR)−θ(m0 TR))]i (9.49)
318CHAPTER9.NOISEANDFREOUENCYCONTROLNote, that the difference between the phases and the timing only depends onthe differencek=m-m'.In the current model phase and timingfluctuationsare uncorrelated. Therefore, for N → oo we obtainS(w) = [ao(w- we)P= eTn(-u),P.(e+j[2(w-wo)(△t(m+k)TR)-t(mTr) (e-j(m+k)TR)-0(mTR)(9.50)Theexpectation values are exactly of the type calculated in (9.43),whichleadstoJao(w -we)PjTR($CE-w)Ke-oo(kTR)S(w) =(9.51)TRe-[(w-we)r)o(kTr)]Most often we are interested in the noise very close to the lines at frequencyoffsets much smaller than the inverse energy and frequency relaxation timesTw and Tp. This is determined by the long term behavior of the variances,which grow linearly in |T元242+0Pn|T| = 2Awe|T],(9.52)ge(T)312w1TPPr([7| = 4A/7](S)72(9.53)t(T)3TWowiththerates27P++0元）1+Awo(9.54)3w1P()Pn2+Awt=(9.55)6TWoFrom the Poisson formula+00+00h[]e-jkN(9.56)>G(r + 2n)
318 CHAPTER 9. NOISE AND FREQUENCY CONTROL Note, that the difference between the phases and the timing only depends on the difference k = m−m0 . In the current model phase and timing fluctuations are uncorrelated. Therefore, for N → ∞ we obtain S(ω) = |aˆ0(ω − ωc)| 2 1 TR P∞ k0=−∞ e jTR ³ ∆φCE TR −ω ´ k he+j[2π(ω−ω0)(∆t((m+k)TR)−∆t(mTR))]i e−j(θ((m+k)TR)−θ(mTR))® . (9.50) The expectation values are exactly of the type calculated in (9.43), which leads to S(ω) = |aˆ0(ω − ωc)| 2 TR X∞ k0=−∞ e jTR ³ φCE TR −ω ´ k e− 1 2 σθ(kTR) (9.51) e−1 2 [((ω−ωc)τ) 2σt(kTR)] Most often we are interested in the noise very close to the lines at frequency offsets much smaller than the inverse energy and frequency relaxation times τ w and τ p. This is determined by the long term behavior of the variances, which grow linearly in |T| σθ(T) = 4 3 µ 1 + π2 12+16 τ 2 w T2 R φ2 o ¶ Pn wo |T| = 2∆ωφ|T|, (9.52) σt(T) = 1 3 Ã π2 + τ 2 p T2 R µD τ 2 ¶2 ! Pn wo |T| = 4∆ωt|T|. (9.53) with the rates ∆ωφ = 2 3 µ 1 + π2 12+16τ 2 w T2 R φ2 o ¶ Pn wo , (9.54) ∆ωt = 1 6 Ã π2 + τ 2 p T2 R µ D τ 2 ¶2 ! Pn wo . (9.55) From the Poisson formula X +∞ k=−∞ h[k]e−jkx = X +∞ n=−∞ G(x + 2nπ) (9.56)