Statistical regimes of random laser fluctuations

Transcription

1 Statistical regimes of random laser fluctuations Stefano Lepri Istituto dei Sistemi Complessi ISC-CNR Firenze S. Cavalieri, G.-L. Oppo, D.S. Wiersma Stefano Lepri (ISC-CNR) Random laser fluctuations 1 / 19

2 An amplifying random medium Stefano Lepri (ISC-CNR) Random laser fluctuations 2 / 19

3 An amplifying random medium Pump beam Stefano Lepri (ISC-CNR) Random laser fluctuations 2 / 19

4 An amplifying random medium Pump beam Stefano Lepri (ISC-CNR) Random laser fluctuations 2 / 19

5 Length scales I L l I 0 Stefano Lepri (ISC-CNR) Random laser fluctuations 3 / 19

6 Length scales I L l Sample size Mean free path Path length Gain length I(l) = I 0 exp(l/l G ) L l l l G I 0 Stefano Lepri (ISC-CNR) Random laser fluctuations 3 / 19

7 Length scales I L l Sample size Mean free path Path length Gain length I(l) = I 0 exp(l/l G ) L l l l G I 0 Diffusive regime, λ l: D = vl/3 If the path is large enough amplification occurs ( mirrorless laser ) Stefano Lepri (ISC-CNR) Random laser fluctuations 3 / 19

8 Spectral narrowing D. S. Wiersma and S. Cavalieri, Nature 414, 708 (2001). Stefano Lepri (ISC-CNR) Random laser fluctuations 4 / 19

9 Fluctuations in emission spectra S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma Phys. Rev. Lett. 93, (2004). Stefano Lepri (ISC-CNR) Random laser fluctuations 5 / 19

10 Statistics of the emitted light Diffusive estimates Distribution of path lengths: p(l) = exp( l/ l ) v DΛ First-passage time estimate: l = Λ smallest eigenvalue of 2 (e.g Λ = q 2, q = π/l for slab of height L) Amplification: I(l) = I 0 exp(l/l G ) Distribution of emitted intensities: l for I > I cutoff. p(i) = l G l I (1+α), α = l G l. Stefano Lepri (ISC-CNR) Random laser fluctuations 6 / 19

11 Statistics of the emitted light Diffusive estimates Distribution of path lengths: p(l) = exp( l/ l ) v DΛ First-passage time estimate: l = Λ smallest eigenvalue of 2 (e.g Λ = q 2, q = π/l for slab of height L) Amplification: I(l) = I 0 exp(l/l G ) Distribution of emitted intensities: l for I > I cutoff. p(i) = l G l I (1+α), α = l G l. Stefano Lepri (ISC-CNR) Random laser fluctuations 6 / 19

12 Statistics of the emitted light Diffusive estimates Distribution of path lengths: p(l) = exp( l/ l ) v DΛ First-passage time estimate: l = Λ smallest eigenvalue of 2 (e.g Λ = q 2, q = π/l for slab of height L) Amplification: I(l) = I 0 exp(l/l G ) Distribution of emitted intensities: l for I > I cutoff. p(i) = l G l I (1+α), α = l G l. Stefano Lepri (ISC-CNR) Random laser fluctuations 6 / 19

13 Statistics of the emitted light Diffusive estimates Distribution of path lengths: p(l) = exp( l/ l ) v DΛ First-passage time estimate: l = Λ smallest eigenvalue of 2 (e.g Λ = q 2, q = π/l for slab of height L) Amplification: I(l) = I 0 exp(l/l G ) Distribution of emitted intensities: l for I > I cutoff. p(i) = l G l I (1+α), α = l G l. Stefano Lepri (ISC-CNR) Random laser fluctuations 6 / 19

14 Statistics of the emitted light Diffusive estimates Distribution of path lengths: p(l) = exp( l/ l ) v DΛ First-passage time estimate: l = Λ smallest eigenvalue of 2 (e.g Λ = q 2, q = π/l for slab of height L) Amplification: I(l) = I 0 exp(l/l G ) Distribution of emitted intensities: l for I > I cutoff. p(i) = l G l I (1+α), α = l G l. Stefano Lepri (ISC-CNR) Random laser fluctuations 6 / 19

15 Gaussian versus Lèvy statistics Detection of the sum of many events, each with distribution p(i) I (1+α) : for α > 2: standard central limit theorem, Gaussian statistics for 0 < α 2, I 2 =! Lèvy statistics; a single large event dominated the overall emission ( lucky photons ) fluctuations do not average out: irreproducible results... dependence on the size, nr. of measurements etc. Stefano Lepri (ISC-CNR) Random laser fluctuations 7 / 19

16 Gaussian versus Lèvy statistics Detection of the sum of many events, each with distribution p(i) I (1+α) : for α > 2: standard central limit theorem, Gaussian statistics for 0 < α 2, I 2 =! Lèvy statistics; a single large event dominated the overall emission ( lucky photons ) fluctuations do not average out: irreproducible results... dependence on the size, nr. of measurements etc. Stefano Lepri (ISC-CNR) Random laser fluctuations 7 / 19

17 Gaussian versus Lèvy statistics Detection of the sum of many events, each with distribution p(i) I (1+α) : for α > 2: standard central limit theorem, Gaussian statistics for 0 < α 2, I 2 =! Lèvy statistics; a single large event dominated the overall emission ( lucky photons ) fluctuations do not average out: irreproducible results... dependence on the size, nr. of measurements etc. Stefano Lepri (ISC-CNR) Random laser fluctuations 7 / 19

18 Gaussian versus Lèvy statistics Detection of the sum of many events, each with distribution p(i) I (1+α) : for α > 2: standard central limit theorem, Gaussian statistics for 0 < α 2, I 2 =! Lèvy statistics; a single large event dominated the overall emission ( lucky photons ) fluctuations do not average out: irreproducible results... dependence on the size, nr. of measurements etc. Stefano Lepri (ISC-CNR) Random laser fluctuations 7 / 19

19 Gaussian versus Lèvy statistics Detection of the sum of many events, each with distribution p(i) I (1+α) : for α > 2: standard central limit theorem, Gaussian statistics for 0 < α 2, I 2 =! Lèvy statistics; a single large event dominated the overall emission ( lucky photons ) fluctuations do not average out: irreproducible results... dependence on the size, nr. of measurements etc. Stefano Lepri (ISC-CNR) Random laser fluctuations 7 / 19

20 Population l G decreases by increasing the population inversion of the active medium. l depends on l and the geometry Thus α = l G l could be made in principle arbitrarily small; But: l G is a time dependent quantity that should be determined from the dynamics of the interacting light-matter system. Need to consider population dynamics in a self consistent way. Stefano Lepri (ISC-CNR) Random laser fluctuations 8 / 19

21 Population l G decreases by increasing the population inversion of the active medium. l depends on l and the geometry Thus α = l G l could be made in principle arbitrarily small; But: l G is a time dependent quantity that should be determined from the dynamics of the interacting light-matter system. Need to consider population dynamics in a self consistent way. Stefano Lepri (ISC-CNR) Random laser fluctuations 8 / 19

22 Population l G decreases by increasing the population inversion of the active medium. l depends on l and the geometry Thus α = l G l could be made in principle arbitrarily small; But: l G is a time dependent quantity that should be determined from the dynamics of the interacting light-matter system. Need to consider population dynamics in a self consistent way. Stefano Lepri (ISC-CNR) Random laser fluctuations 8 / 19

23 Population l G decreases by increasing the population inversion of the active medium. l depends on l and the geometry Thus α = l G l could be made in principle arbitrarily small; But: l G is a time dependent quantity that should be determined from the dynamics of the interacting light-matter system. Need to consider population dynamics in a self consistent way. Stefano Lepri (ISC-CNR) Random laser fluctuations 8 / 19

24 Population l G decreases by increasing the population inversion of the active medium. l depends on l and the geometry Thus α = l G l could be made in principle arbitrarily small; But: l G is a time dependent quantity that should be determined from the dynamics of the interacting light-matter system. Need to consider population dynamics in a self consistent way. Stefano Lepri (ISC-CNR) Random laser fluctuations 8 / 19

25 Population l G decreases by increasing the population inversion of the active medium. l depends on l and the geometry Thus α = l G l could be made in principle arbitrarily small; But: l G is a time dependent quantity that should be determined from the dynamics of the interacting light-matter system. Need to consider population dynamics in a self consistent way. Stefano Lepri (ISC-CNR) Random laser fluctuations 8 / 19

26 What should we expect? Time scales: Gain larger then losses : r = v/l G DΛ > 0 [V.S. Letokhov Sov. Phys. JETP 26, 835 (1968)] Overall intensity grows initially as exp(rt); Short pump pulses: population is strongly depleted (l G becomes large) on time 1/r; The residence time l /v must thus be smaller than 1/r for large events to be observed; Conditions for the Lévy regime: 1/r > l /v and α 2: 1 v 2 DΛ < l G < 2 v DΛ. The lower bound correspond to the minimal value α = 1/2. Stefano Lepri (ISC-CNR) Random laser fluctuations 9 / 19

27 What should we expect? Time scales: Gain larger then losses : r = v/l G DΛ > 0 [V.S. Letokhov Sov. Phys. JETP 26, 835 (1968)] Overall intensity grows initially as exp(rt); Short pump pulses: population is strongly depleted (l G becomes large) on time 1/r; The residence time l /v must thus be smaller than 1/r for large events to be observed; Conditions for the Lévy regime: 1/r > l /v and α 2: 1 v 2 DΛ < l G < 2 v DΛ. The lower bound correspond to the minimal value α = 1/2. Stefano Lepri (ISC-CNR) Random laser fluctuations 9 / 19

28 What should we expect? Time scales: Gain larger then losses : r = v/l G DΛ > 0 [V.S. Letokhov Sov. Phys. JETP 26, 835 (1968)] Overall intensity grows initially as exp(rt); Short pump pulses: population is strongly depleted (l G becomes large) on time 1/r; The residence time l /v must thus be smaller than 1/r for large events to be observed; Conditions for the Lévy regime: 1/r > l /v and α 2: 1 v 2 DΛ < l G < 2 v DΛ. The lower bound correspond to the minimal value α = 1/2. Stefano Lepri (ISC-CNR) Random laser fluctuations 9 / 19

29 What should we expect? Time scales: Gain larger then losses : r = v/l G DΛ > 0 [V.S. Letokhov Sov. Phys. JETP 26, 835 (1968)] Overall intensity grows initially as exp(rt); Short pump pulses: population is strongly depleted (l G becomes large) on time 1/r; The residence time l /v must thus be smaller than 1/r for large events to be observed; Conditions for the Lévy regime: 1/r > l /v and α 2: 1 v 2 DΛ < l G < 2 v DΛ. The lower bound correspond to the minimal value α = 1/2. Stefano Lepri (ISC-CNR) Random laser fluctuations 9 / 19

30 What should we expect? Time scales: Gain larger then losses : r = v/l G DΛ > 0 [V.S. Letokhov Sov. Phys. JETP 26, 835 (1968)] Overall intensity grows initially as exp(rt); Short pump pulses: population is strongly depleted (l G becomes large) on time 1/r; The residence time l /v must thus be smaller than 1/r for large events to be observed; Conditions for the Lévy regime: 1/r > l /v and α 2: 1 v 2 DΛ < l G < 2 v DΛ. The lower bound correspond to the minimal value α = 1/2. Stefano Lepri (ISC-CNR) Random laser fluctuations 9 / 19

31 What should we expect? Time scales: Gain larger then losses : r = v/l G DΛ > 0 [V.S. Letokhov Sov. Phys. JETP 26, 835 (1968)] Overall intensity grows initially as exp(rt); Short pump pulses: population is strongly depleted (l G becomes large) on time 1/r; The residence time l /v must thus be smaller than 1/r for large events to be observed; Conditions for the Lévy regime: 1/r > l /v and α 2: 1 v 2 DΛ < l G < 2 v DΛ. The lower bound correspond to the minimal value α = 1/2. Stefano Lepri (ISC-CNR) Random laser fluctuations 9 / 19

32 (L, l G ) diagram 600 2D slab of thickness L, l = 1, v = 1 l G Gaussian Laser threshold Levy α=2 α=1 α=1/2 Gaussian L Stefano Lepri (ISC-CNR) Random laser fluctuations 10 / 19

33 Model (I) r = (x, y) x L l l y RL Cells of size l, periodic b.c. along y Stefano Lepri (ISC-CNR) Random laser fluctuations 11 / 19

34 Model (I) x L l l N(r, t) y RL Hypothetical three level system with N 1 0, N 2 N Stefano Lepri (ISC-CNR) Random laser fluctuations 11 / 19

35 Model (I) L n i (t + 2 t) n i (t + t) x n i (t) y RL t = l/v, walkers labeled by i = 1,..., M Stefano Lepri (ISC-CNR) Random laser fluctuations 11 / 19

36 Model (II) 1 Pumping: let N(r, t = 0) = N 0, initial number of walkers M = 0. 2 Spontaneous emission: For each t and r and probability γn t (γ = sp. em. rate of the single atom) start a new walker with energy n ini, then N N 1, M M Diffusion: Parallel and asynchronous update of the walkers positions: random move to one of the 4 neighbouring cells. If the boundaries x = 1, L are reached, record n out and let M M 1. 4 Stimulated emission: For i = 1,..., M apply deterministic rule n i (1 + γ t N) n i, N (1 γ t n i ) N, N is the population at the site where i th walker resides. Stefano Lepri (ISC-CNR) Random laser fluctuations 12 / 19

37 Model (II) 1 Pumping: let N(r, t = 0) = N 0, initial number of walkers M = 0. 2 Spontaneous emission: For each t and r and probability γn t (γ = sp. em. rate of the single atom) start a new walker with energy n ini, then N N 1, M M Diffusion: Parallel and asynchronous update of the walkers positions: random move to one of the 4 neighbouring cells. If the boundaries x = 1, L are reached, record n out and let M M 1. 4 Stimulated emission: For i = 1,..., M apply deterministic rule n i (1 + γ t N) n i, N (1 γ t n i ) N, N is the population at the site where i th walker resides. Stefano Lepri (ISC-CNR) Random laser fluctuations 12 / 19

38 Model (II) 1 Pumping: let N(r, t = 0) = N 0, initial number of walkers M = 0. 2 Spontaneous emission: For each t and r and probability γn t (γ = sp. em. rate of the single atom) start a new walker with energy n ini, then N N 1, M M Diffusion: Parallel and asynchronous update of the walkers positions: random move to one of the 4 neighbouring cells. If the boundaries x = 1, L are reached, record n out and let M M 1. 4 Stimulated emission: For i = 1,..., M apply deterministic rule n i (1 + γ t N) n i, N (1 γ t n i ) N, N is the population at the site where i th walker resides. Stefano Lepri (ISC-CNR) Random laser fluctuations 12 / 19

39 Model (II) 1 Pumping: let N(r, t = 0) = N 0, initial number of walkers M = 0. 2 Spontaneous emission: For each t and r and probability γn t (γ = sp. em. rate of the single atom) start a new walker with energy n ini, then N N 1, M M Diffusion: Parallel and asynchronous update of the walkers positions: random move to one of the 4 neighbouring cells. If the boundaries x = 1, L are reached, record n out and let M M 1. 4 Stimulated emission: For i = 1,..., M apply deterministic rule n i (1 + γ t N) n i, N (1 γ t n i ) N, N is the population at the site where i th walker resides. Stefano Lepri (ISC-CNR) Random laser fluctuations 12 / 19

40 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

41 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

42 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

43 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

44 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

45 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

46 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

47 Remarks Stochasticity is both in the spontaneous emission (Step 2) and random walk (Step 3); Monte-Carlo run: iterate Steps 2 4 up to a preassigned time; Record all the outflowing photon numbers n out, average over sample surface and subsequent time windows [t, t + T W ]: time-series of the flux φ(t); Units: v = 1, l = 1 (and thus t = 1); free parameters γ, N 0, L, RL; Make it simple : we have neglected many things... non radiative decay γ = γ(λ) homogeneous and istantaneous pumping process Stefano Lepri (ISC-CNR) Random laser fluctuations 13 / 19

48 Mean-field equations Rate equations for the macroscopic averages Ṅ = γn(i + 1) I = D I + γn(i + 1) I(r, t)=nr. of photons in each cell, = 2D discrete Laplacian. Absorbing b.c. I(0, t) = I(L + 1, t) = 0, N(x, 0) = N 0, I(x, 0) = 0 [L. Florescu and S. John, Phys. Rev. E (2004)] Threshold condition: where N c = Dq 2 /γ, q = π/l. r = γn 0 Dq 2 = γ(n 0 N c ) > 0, Stefano Lepri (ISC-CNR) Random laser fluctuations 14 / 19

49 Mean-field equations Rate equations for the macroscopic averages Ṅ = γn(i + 1) I = D I + γn(i + 1) I(r, t)=nr. of photons in each cell, = 2D discrete Laplacian. Absorbing b.c. I(0, t) = I(L + 1, t) = 0, N(x, 0) = N 0, I(x, 0) = 0 [L. Florescu and S. John, Phys. Rev. E (2004)] Threshold condition: where N c = Dq 2 /γ, q = π/l. r = γn 0 Dq 2 = γ(n 0 N c ) > 0, Stefano Lepri (ISC-CNR) Random laser fluctuations 14 / 19

50 Output pulse 10 2 (a) N 0 = < N c: Subthreshold Lèvy photon flux φ time L = 30, R = 20, γ = 10 12, T W = 10, N c = Stefano Lepri (ISC-CNR) Random laser fluctuations 15 / 19

51 Output pulse N 0 = N c: Suprathreshold Lèvy 10 8 (b) 10 6 photon flux φ time L = 30, R = 20, γ = 10 12, T W = 10, N c = Stefano Lepri (ISC-CNR) Random laser fluctuations 15 / 19

52 Output pulse (c) N 0 = > N c: Gaussian 10 8 photon flux φ time L = 30, R = 20, γ = 10 12, T W = 10, N c = Stefano Lepri (ISC-CNR) Random laser fluctuations 15 / 19

53 Volume-averaged population normalized average population time Stefano Lepri (ISC-CNR) Random laser fluctuations 16 / 19

54 Lèvy region: distribution of N(r, t) N 0 N c: Suprathreshold Lèvy, t = 10 5 x L y Stefano Lepri (ISC-CNR) Random laser fluctuations 17 / 19

55 Lèvy region: distribution of n out N 0 =2 x 10 9 N 0 =N C N 0 =5 x 10 9 p photon number Stefano Lepri (ISC-CNR) Random laser fluctuations 18 / 19

56 Lèvy region: distribution of n out N 0 =2 x 10 9 N 0 =N C N 0 =5 x 10 9 p α l G photon number Stefano Lepri (ISC-CNR) Random laser fluctuations 18 / 19

57 Outlook Random amplifying media; A statistical explanation for fluctuations in the emission spectra; Role of population inversion; Gaussian and Lèvy regimes depending on DΛl G /v; A Monte-Carlo model. S. L., S. Cavalieri, G-L. Oppo, D. S. Wiersma, Phys. Rev. A 75, (2007) Stefano Lepri (ISC-CNR) Random laser fluctuations 19 / 19

58 Outlook Random amplifying media; A statistical explanation for fluctuations in the emission spectra; Role of population inversion; Gaussian and Lèvy regimes depending on DΛl G /v; A Monte-Carlo model. S. L., S. Cavalieri, G-L. Oppo, D. S. Wiersma, Phys. Rev. A 75, (2007) Stefano Lepri (ISC-CNR) Random laser fluctuations 19 / 19