Mode-locked laser pulse fluctuations

Transcription

1 Mode-locked laser pulse fluctuations Omri Gat, Racah Institute of Physics, Hebrew University Joint work with: Michael Katz, Baruch Fischer Acknowledgement: Rafi Weill, Oded Basis, Alex Bekker, Vladimir Smulakovsky Supported by: Israel Science Foundation

2 Synopsis Mode-locked soliton lasers and noise The statistical steady state Fluctuations in the steady state 1. Pulse-continuum interactions 2. Gain fluctuations 3. Slow modes of pulse dynamics 4. Pulse parameter equations of motion Autocorrelation and diffusion of pulse parameters

3 Mode-locked soliton lasers dispersive medium Output

4 Mode-locked soliton lasers dispersive medium Output Ultrashort light pulses <1ps high intensity, broad bandwidth

5 Mode-locked soliton lasers dispersive medium Output Ultrashort light pulses <1ps high intensity, broad bandwidth Dominant dispersive effects: Chromatic dispersion ( anomalous ) linear Kerr effect nonlinear

6 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Pulse shaping effects: Overall gain & gain filtering linear Saturable absorption : Intensity-bleached absorbing element nonlinear Relatively weak: proportional to µ 1

7 Mode-locked soliton lasers dispersive medium Pump Output [Haus 75] saturable absorber Master equation of motion for the field envelope 1 z ψ =(i + µ) 2 2 t ψ + ψ 2 ψ + gψ ψ dispersive effects (non-dimensionalized) pulse shaping overall net gain (<0)

8 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) Soliton-like pulse: Parameters: a = 1 2 P t ψ + ψ 2 ψ + gψ ψ s (t, z) =asech( t C(z) τ C = c Vdz τ = 1 a Φ = φ Vt )e iφ(t,z) ψ (P 2 + V 2 )dz

9 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) Soliton-like pulse: Parameters: a = 1 2 P τ = 1 a t ψ + ψ 2 ψ + gψ ψ s (t, z) =asech( t C(z) τ C = c Vdz Φ = φ Vt )e iφ(t,z) ψ (P 2 + V 2 )dz Timing Phase Power Frequency

10 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) Soliton-like pulse: Parameters: a = 1 2 P τ = 1 a t ψ + ψ 2 ψ + gψ ψ s (t, z) =asech( t C(z) τ C = c Vdz Φ = φ Vt )e iφ(t,z) ψ (P 2 + V 2 )dz Timing Phase Power Frequency

11 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) Soliton-like pulse: Parameters: a = 1 2 P τ = 1 a t ψ + ψ 2 ψ + gψ ψ s (t, z) =asech( t C(z) τ C = c Vdz Φ = φ Vt )e iφ(t,z) ψ (P 2 + V 2 )dz Timing Phase Power Frequency

12 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) Soliton-like pulse: Parameters: a = 1 2 P τ = 1 a t ψ + ψ 2 ψ + gψ ψ s (t, z) =asech( t C(z) τ C = c Vdz Φ = φ Vt )e iφ(t,z) ψ (P 2 + V 2 )dz Timing Phase Power Frequency

13 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) t ψ + ψ 2 ψ Pulse parameters: Power P = 8 g & frequency V = 0 fixed µ Pulse shaping: Singular perturbation + gψ Timing c and phase φ free exact symmetries ψ

14 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) t ψ + ψ 2 ψ Pulse parameters: Power P = 8 g & frequency V = 0 fixed µ Pulse shaping: Singular perturbation + gψ Timing c and phase φ free exact symmetries ψ

15 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) t ψ + ψ 2 ψ Pulse parameters: Power P = 8 g & frequency V = 0 fixed µ Pulse shaping: Singular perturbation + gψ Timing c and phase φ free exact symmetries ψ

16 Mode-locked soliton lasers dispersive medium Pump Output saturable absorber Master equation of motion for the field envelope z ψ =(i + µ) t ψ + ψ 2 ψ Ideal output: Periodic pulse train + gψ ψ Output intensity Time

17 Noise and fluctuations Amplification & noise Output quasi-cw background quasi-cw background Noisy master equation: 1 z ψ =(i + µ) 2 2 t ψ + ψ 2 ψ + gψ + Γ(z, t) Weak Gaussian white noise Γ(z 1, t 1 )Γ (z 2, t 2 ) = 2 2 TLδ(z 1 z 2 )δ(t 1 t 2 ) 1 Noise power injection rate

18 Noise and fluctuations Amplification & noise Output quasi-cw background quasi-cw background Noisy master equation: 1 z ψ =(i + µ) 2 2 t ψ + ψ 2 ψ + gψ + Γ(z, t) Weak Gaussian white noise Γ(z 1, t 1 )Γ (z 2, t 2 ) = 2 2 TLδ(z 1 z 2 )δ(t 1 t 2 ) 1 Waveform is perturbed ψ = ψ s (t, z)+o() Noise power injection rate

19 Statistical light-mode dynamics Amplification & noise Output quasi-cw background quasi-cw background Perturbed waveform: ψ = ψ s (t, z)+o() Ordered pulse waveform Noisy waveform Output intensity quasi-cw background pulse fluctuations Mode phasors Time

20 Statistical light-mode dynamics Amplification & noise Output quasi-cw background quasi-cw background Perturbed waveform: 1. Pulse fluctuations ψ = ψ s (t, z)+o() Ordered pulse waveform Noisy waveform Output intensity quasi-cw background pulse fluctuations Mode phasors Time

21 Statistical light-mode dynamics Amplification & noise Output quasi-cw background quasi-cw background Perturbed waveform: 1. Pulse fluctuations ψ = ψ s (t, z)+o() Ordered pulse waveform Noisy waveform 2. Low-intensity quasi-cw background Mode phasors Output intensity quasi-cw background pulse fluctuations Time

22 Statistical light-mode dynamics Amplification & noise Output quasi-cw background quasi-cw background Pulse waveform ψ s + ψ 1 : Strong & narrow Continuum waveform ψ c : Weak & wide Output intensity quasi-cw background Mode phasors Time

23 Statistical light-mode dynamics Amplification & noise Output quasi-cw background quasi-cw background Pulse waveform ψ s + ψ 1 : Strong & narrow Continuum waveform ψ c : Weak & wide Pulse power ~ Continuum power ~ total power Mode phasors Output intensity quasi-cw background Time

24 Statistical light-mode dynamics Amplification & noise Output quasi-cw background quasi-cw background Pulse waveform ψ s + ψ 1 : Strong & narrow Continuum waveform ψ c : Weak & wide Pulse power ~ Continuum power ~ total power For strong noise, disordering first order transition to cw phase Output intensity Mode phasors Time

25 SLD: The gain balance method Amplification & noise Output quasi-cw background quasi-cw background ψ s + ψ 1 ψ c P P

26 SLD: The gain balance method Amplification & noise Output quasi-cw background quasi-cw background Pulse waveform ψ s + ψ 1 : Strong & narrow Neglect noise Continuum waveform ψ c : Weak & wide Neglect nonlinearity Pulse power P + continuum power = total power P

27 SLD: The gain balance method Amplification & noise Output quasi-cw background quasi-cw background Pulse waveform ψ s + ψ 1 : Strong & narrow Neglect noise Continuum waveform ψ c : Weak & wide Neglect nonlinearity Pulse power P + continuum power = total power P Net gain g Common gain value determines the power distribution between pulse & continuum

28 SLD: The gain balance method Steady state power distribution P P = L 2 µ 8T P 2 Thermodynamic limit L 1 µ 1 Continuum gain Pulse gain Pulse waveform ψ s + ψ 1 : Strong & narrow Neglect noise Continuum waveform ψ c : Weak & wide Neglect nonlinearity Pulse power P + continuum power = total power P P P Net gain g Mode locking is possible only if a consistent power distribution between pulse & continuum exists

29 Pulse fluctuations Noise Noise causes jitter in pulse parametrs Power P & frequency V fluctuate Timing c and phase φ diffuse

30 Pulse fluctuations [Haus & Mecozzi 93] Noise Question: What are the statistical properties of the fluctuations? Practical implications: Performance of pulse sources Precision of frequency-comb metrology

31 SLD beyond thermodynamics Idea: Decompose wave form in 3 parts strong & narrow weak & wide weak & narrow ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) fluctuating waveform, power = P soliton-like pulse with fluctuating parameter linear continuum, power ~ P P(z) c(z), φ(z), P(z), V(z) overlap waveform power 1 1

32 SLD beyond thermodynamics Idea: Decompose wave form in 3 parts strong & narrow weak & wide weak & narrow ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) fluctuating waveform, power = P soliton-like pulse with fluctuating parameter linear continuum, power ~ P P(z) c(z), φ(z), P(z), V(z) Main goal: Calculate statistics of pulse paramaters overlap waveform power 1 1 Negligible in steady-state Important for fluctuations

33 SLD beyond thermodynamics ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) c(z), φ(z), P(z), V(z) New qualitative effects: 1. Oscillations in power & frequency correlation functions 2. Enhancement of phase diffusion rate

34 1. The perturbation equation ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Let P = P 0 + µ p(z), V = steady-state pulse power µ v(z), c µ c(z), φ fluctuations small parameter µ φ(z)

35 1. The perturbation equation ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Let P = P 0 + µ p(z), V = steady-state pulse power µ v(z), c µ c(z), φ fluctuations small parameter µ φ(z) Linearized master equation for z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + g 1 ψ s + f parameter set ψ 1 gain fluctuations Nonlinearity-generated forcing by overlap of ψ s & ψ c

36 2. Gain fluctuations ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Double role of net gain in the steady state: 1. Determines the pulse & continuum power g 0 = µ 8 P 0 Gain balance 2. Determined by the overall power g 0 = g(p) Gain saturation

37 2. Gain fluctuations ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Double role of net gain in the steady state: 1. Determines the pulse & continuum power g 0 = µ 8 P 0 Gain balance 2. Determined by the overall power g 0 = g(p) Gain saturation Gain fluctuations g = g 0 + g 1 arise from 1. Random shuffle of power between pulse and continuum 2. Fluctuations of overall power

38 2. Gain fluctuations ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Double role of net gain in the steady state: 1. Determines the pulse & continuum power g 0 = µ 8 P 0 Gain balance 2. Determined by the overall power g 0 = g(p) Gain saturation Gain fluctuations g = g 0 + g 1 arise from 1. Random shuffle of power between pulse and continuum 2. Fluctuations of overall power Assume deep saturation

39 2. Gain fluctuations ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Gain fluctuations g = g 0 + g 1 arise from 1. Random shuffle of power between pulse and continuum 2. Fluctuations of overall power Linearized master equation for Assume deep saturation ψ 1 z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + g 1 ψ s + f

40 2. Gain fluctuations ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Gain fluctuations g = g 0 + g 1 arise from 1. Random shuffle of power between pulse and continuum 2. Fluctuations of overall power Linearized master equation for ψ 1 Fluctuation conserve overall power g 1 P = µp p dzψ s Γ + L(ψc + ψ 1 ) Assume deep saturation z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + g 1 ψ s + f

41 3. The slow modes ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Perturbation equation including gain fluctuations: z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + f Linear operator including rank-1 gain fluctuations term Forcing including gain fluctuations

42 3. The slow modes ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Perturbation equation including gain fluctuations: z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + f Linear operator including rank-1 gain fluctuations term Remaining arbitrariness: A slight shift in absorbed in, Forcing including gain fluctuations can be x ψ 1 ψ s (x)+ψ 1 = ψ s (x + δx)+(ψ 1 + δψ 1 )

43 3. The slow modes ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Perturbation equation including gain fluctuations: z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + f Linear operator including rank-1 gain fluctuations term Remaining arbitrariness: A slight shift in absorbed in, Forcing including gain fluctuations can be x ψ 1 ψ s (x)+ψ 1 = ψ s (x + δx)+(ψ 1 + δψ 1 ) Q: How to define the pulse parameters? A: Let ψ 1 lie outside 4-dimensional slow eigen-space of L

44 3. The slow modes ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Perturbation equation including gain fluctuations: z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + f Recall: linearized NLS has 4-d zero eigen-space: timing, phase: annihilated by L NLS frequency, power: annihilated by L 2 NLS

45 3. The slow modes ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Perturbation equation including gain fluctuations: z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + f Recall: linearized NLS has 4-d zero eigen-space: timing, phase: annihilated by L NLS frequency, power: annihilated by L 2 NLS Degeneracy is half-lifted by O(µ) L = L NLS + O(µ) timing, phase: annihilated by L O(µ). eigenfunctions of L

46 4. Pulse parameter dynamics ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Perturbation equation: where: z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + f ψ 1 lies outside 4-dimensional slow eigen-space of where are slow left eigenfunctions q n, ψ 1 = 0 q 1,...,q 4 L L

47 4. Pulse parameter dynamics ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) Perturbation equation: where: z ψ µ z x x ψ s = Lψ 1 i 1 2 µv(z) t ψ s + f ψ 1 lies outside 4-dimensional slow eigen-space of where are slow left eigenfunctions q n, ψ 1 = 0 q 1,...,q 4 L L q n, projection of the perturbation equation yields (linear combination of) parameter equations of motion

48 4. Pulse parameter dynamics ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) q n, projection of the perturbation equation yields (linear combination of) parameter equations of motion: Power: Phase: Frequency: Timing: z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = µ fc Restoring terms Random forcing

49 4. Pulse parameter dynamics ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) q n, projection of the perturbation equation yields (linear combination of) parameter equations of motion: Power: Phase: Frequency: Timing: z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = µ fc Bounded fluctuations Restoring terms Random forcing

50 4. Pulse parameter dynamics ψ(t, z) = ψ s (t, z) + ψ c (t, z) + ψ 1 (t, z) q n, projection of the perturbation equation yields (linear combination of) parameter equations of motion: Power: Phase: Frequency: Timing: z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = µ fc Bounded fluctuations Diffusion Restoring terms Random forcing

51 Results: Autocorrelation Pulse frequency: v t+τ v t = T e µp2 0 τ 6 + π P 0 dkk 2 I k (τ) Parameter fluctuations Direct term: exponential damping

52 Results: Autocorrelation Pulse frequency: v t+τ v t = T e µp2 0 τ 6 + π P 0 dkk 2 I k (τ) Parameter fluctuations I k (τ) = sech2 (πk/2) k 2 +1 Direct term: exponential damping Continuum term: Damped oscillations e µp2 0 (k2 +1) 8 τ cos( P2 0 (k2 +1) 8 τ)

53 Results: Autocorrelation Pulse frequency: v t+τ v t = T e µp2 0 τ 6 + π P 0 dkk 2 I k (τ) Parameter fluctuations I k (τ) = sech2 (πk/2) k 2 +1 Direct term: exponential damping Continuum term: Damped oscillations e µp2 0 (k2 +1) 8 τ cos( P2 0 (k2 +1) 8 τ) Pulse power: p t+τ p t = 2T P 0 PP0 e µp3 0 4P τ + π 2 Continuum enhancement factor dkik (τ)

54 Results: Autocorrelation Pulse frequency: v t+τ v t = T e µp2 0 τ 6 + π P 0 dkk 2 I k (τ) frequency power Direct term: exponential damping Continuum term: Damped oscillations Pulse power: p t+τ p t = 2T P 0 PP0 e µp3 0 4P τ + π 2 Continuum enhancement factor dkik (τ)

55 Results: Autocorrelation Pulse frequency: v t+τ v t = T e µp2 0 τ 6 + π P 0 dkk 2 I k (τ) power frequency power Theory Numerics Alternative parameter definition frequency Pulse power: p t+τ p t = 2T P 0 PP0 e µp3 0 4P τ + π 2 dkik (τ) [Menyuk & al] oscillations generated by gain dynamics: Here suppressed

56 Results: Autocorrelation Pulse frequency: v t+τ v t = T e µp2 0 τ 6 + π P 0 dkk 2 I k (τ) power frequency power Theory Numerics Alternative parameter definition frequency Pulse power: p t+τ p t = 2T P 0 PP0 e µp3 0 4P τ + π 2 dkik (τ) [Menyuk & al] oscillations generated by gain dynamics: Here suppressed

57 Results: Autocorrelation Pulse frequency: Pulse power: Continuum interaction: I k (τ) = sech2 (πk/2) k 2 +1 v t+τ v t = T e µp2 0 τ 6 + π P 0 p t+τ p t = 2T P e µp 0 3 4P τ + π P 0 P 0 2 Nonlinear phase shift dkk 2 I k (τ) e µp2 0 8 (k 2 +1) τ cos( P2 0 8 (k 2 + 1)τ) Pulse shaping rate dki k (τ)

58 Results: Autocorrelation Pulse frequency: Pulse power: Continuum interaction: I k (τ) = sech2 (πk/2) k 2 +1 Compare with pulse-generated continuum: Kelly sidebands v t+τ v t = T e µp2 0 τ 6 + π P 0 p t+τ p t = 2T P e µp 0 3 4P τ + π P 0 P 0 2 Nonlinear phase shift dkk 2 I k (τ) e µp2 0 8 (k 2 +1) τ cos( P2 0 8 (k 2 + 1)τ) Pulse shaping rate dki k (τ)

59 Parameter equations z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = µ fc Results: Diffusion

60 Results: Diffusion Parameter equations Pulse timing & phase z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = C = c Vdz µ fc Φ = φ Vt Direct diffusion (P 2 + V 2 )dz

61 Results: Diffusion Parameter equations Pulse timing & phase z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = C = c Vdz µ fc Φ = φ Vt Direct diffusion (P 2 + V 2 )dz Frequency- & powerfluctuations driven diffusion

62 Results: Diffusion Parameter equations Pulse timing & phase z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = C = c Vdz µ fc Φ = φ Vt Direct diffusion (P 2 + V 2 )dz Frequency- & powerfluctuations driven diffusion Driven diffusion Direct diffusion Damped-term driven diffusion Oscillatory-term driven

63 Results: Diffusion Parameter equations Pulse timing & phase z p = µ P3 0 4P p µ f p z φ = µ fφ z v = µ P2 0 6 v µ f v z c = C = c Vdz µ fc Φ = φ Vt Direct diffusion (P 2 + V 2 )dz Frequency- & powerfluctuations driven diffusion Driven diffusion Direct diffusion Damped-term driven diffusion Oscillatory-term driven Timing jitter Phase jitter C(z) 2 = 12 T z P0 3 Φ(z) 2 = TP2 z P0 3 Diffusion constants = Haus-Mecozzi jitter enhanced by P P 0 2

64 Summary & conclusions

65 Summary & conclusions Laser noise has dual effect 1. Steady-state linear continuum 2. Pulse fluctuations

66 Summary & conclusions Laser noise has dual effect 1. Steady-state linear continuum 2. Pulse fluctuations Fluctuations in the statistical steady state: 3-part decomposition of the light waveform

67 Summary & conclusions Laser noise has dual effect 1. Steady-state linear continuum 2. Pulse fluctuations Fluctuations in the statistical steady state: 3-part decomposition of the light waveform Arbitrariness in pulse parameter values: removed by slow-modes projection

68 Summary & conclusions Laser noise has dual effect 1. Steady-state linear continuum 2. Pulse fluctuations Fluctuations in the statistical steady state: 3-part decomposition of the light waveform Arbitrariness in pulse parameter values: removed by slow-modes projection Autocorrelation oscillations caused by continuum-pulse interaction (cf. Kelly sidebands)

69 Summary & conclusions Laser noise has dual effect 1. Steady-state linear continuum 2. Pulse fluctuations Fluctuations in the statistical steady state: 3-part decomposition of the light waveform Arbitrariness in pulse parameter values: removed by slow-modes projection Autocorrelation oscillations caused by continuum-pulse interaction (cf. Kelly sidebands) Restoring terms suppressed by continuum: enhanced phase jitter

70 Outlook Laser noise has triple effect 1. Steady-state linear continuum 2. Pulse fluctuations 3. Pulse interactions

71 Outlook Laser noise has triple effect 1. Steady-state linear continuum 2. Pulse fluctuations 3. Pulse interactions Thank you!