Laser Systems and Applications

Transcription

1 MSc in Photonics & Europhotonics Laser Systems and Applications Cristina Masoller Research group on Dynamics, onlinear Optics and Lasers (DOLL) Departament de Física i Enginyeria uclear Universitat Politècnica de Catalunya cristina.masoller@upc.edu

2 Outline Block 1: Low power laser systems Introduction Semiconductor light sources Models Dynamical effects Applications: telecom & datacom, storage, others

3 Goals Acquire a basic knowledge of the rate equation model for the photon and carrier densities. Steady-state solutions and LI curve. Laser dynamics with time-varying injection current Turn on delay and relaxation oscillations Dynamic LI curve and hysteresis Current modulation and modulation bandwidth Become familiar with the rate equation for the complex optical field. Phase-amplitude coupling Laser line-width and intensity noise Understand optical injection, feedback & polarization effects 06/01/2016 3

4 Outline: Models and dynamical effects Simplest rate equation model: photon and carrier density Steady state solutions Time dependent solutions (turn on, relaxation oscillations, response to periodic modulation) Rate equation model for the complex optical field Response to optical perturbations Polarization dynamics 06/01/2016 4

5 Semiconductor lasers are class B lasers Governed by two rate-equations: one for the photons (S) and one for the carriers (). Display a stable output (with only transient relaxation oscillations). Single-mode conventional EELs diode lasers are class B lasers. Other class B lasers are ruby, d:yag, and CO2 lasers. Because of the -factor (a specific feature of diode lasers, more latter) diode lasers display complex dynamics when they are optically perturbed. n = Carrier lifetime p = Photon lifetime Characteristic times for common class B lasers 06/01/2016 5

6 Other types of lasers Class A (Visible He-e lasers, Ar-ion lasers, dye lasers): governed by one rate equation for the optical field (the material variables can be adiabatically eliminated), no oscillations. Class C (He-e lasers operating at infrared lines): governed by three rate equations (, S, P=macroscopic atomic polarization), display sustained oscillations and even a chaotic output. o commercial applications. 06/01/2016 6

7 Diode lasers : electrical to power conversion Injected electrical current Carrier density et gain: emission absorption Photon density Cavity losses 06/01/2016 7

8 Rate equation for the carrier density d dt I ev GS Pump: injection of carriers Recombination of carriers Stimulated emission & absorption I : Injection current (I/eV is the number of electrons injected per unit volume and per unit time). : Carrier lifetime. G (,S) : et gain 06/01/2016 8

9 Rate equation for the photon density S ds dt GS S p sp Stimulated emission - absorption Cavity losses Spontaneous emission p : Photon lifetime. G (,S) : et gain sp : Spontaneous emission rate 06/01/2016 9

10 Simple model for the semiconductor gain RT InGaASP laser G a G a 0 0 ln / 0 DH QW Differential gain coefficient Carrier density at transparency 0 We assume singlemode emission at 0. The differential gain coefficient a depends on 0 (multi-mode model latter). 06/01/

11 Threshold carrier density Threshold condition: net gain = cavity loss G( ) th 1 p G a 0 1 p a th 0 G ds dt 1 GS S p sp a a a 0 th 0 p th ds dt a th S sp 06/01/

12 Two coupled nonlinear rate-equations ds dt GS S p sp d dt I ev GS Ordinary differential equations (spatial effects neglected) Additional nonlinearities from carrier re-combination and gain saturation 1 A B C These equations allow to understand the LI curve and the modulation response. 2 To understand the intensity noise and the line-width (the optical spectrum), we need a stochastic equation for the complex field E (S= E 2 ) (more latter). Spatial effects (diffraction, carrier diffusion) and thermal effects can be included phenomenologically. G a 1 S 0 06/01/

13 ormalized equations Define the dimensionless variable: ds dt 1 1 p d 1 dt S sp S Pump current parameter: proportional to I/I th 0 th Threshold carrier density: gain = loss a th 0 1 p ormalizing the equations eliminates two parameters (a, o ) In the following we drop the 06/01/

14 Role of spontaneous emission ds dt 1 1 p S sp If at t=0 there are no photons in the cavity: S(0) = 0 Then, without noise ( sp =0): if S=0 at t=0 ds/dt=0 S remains 0 (regardless the value of and ). Without spontaneous emission noise the laser does not turn. 06/01/

15 Steady state solutions with sp =0 (Simple expressions if sp is neglected) ds dt ds dt 1 1S S p S d dt d dt 1 S S 1 Laser off Laser on Stable if <1 S=0 = Stable if >1 S = -1 =1 th = 1 Above threshold the carriers are clamped. 06/01/

16 S Graphical representation Pump I Pump I Pump current, Pump current, th = 1 off on S=0 S=-1 off on Above threshold the carriers are clamped. = = 1 06/01/

17 Experimental LI curve Hitachi Laser Diode HL6724MG (A. Aragoneses PhD thesis 2014) ds dt a th S sp This LI curve is obtained from model simulations when sp is not neglected. 06/01/

18 LI curve with sp 0 ds dt d dt 1 1 p 1 S sp S p b t t' sp p p ds dt' d dt' S b 1 S Steady-state solution: b S 4 2 If >1 S -1 If <1 S 1/ -1 06/01/

19 kink of the LI curve If >1 S -1 If <1 S 1/ -1 th = 1 GaAs diode laser (1982) From T. Erneux and P. Glorieux, Laser Dynamics (2010) 06/01/

20 Model for a multi-mode laser Gain coefficient for mode j: Carrier density (n) + several photon densities (for each longitudinal mode) 06/01/

21 SIMPLEST RATE EQUATIO MODEL -STEADY-STATE SOLUTIOS & LI CURVE -TIME-DEPEDET SOLUTIOS WHE THE IJECTIO CURRET VARIES 06/01/

22 Time variation of the injection current d dt 1 S ds dt 1 1 p S sp (t) Step (laser turn on): off, on Parameter values p 1 ps 1 ns sp 10-4 Triangular (dynamic LI curve): min, max, T ramp Sinusoidal (modulation response): dc, A, T mod 06/01/

23 Current step: turn-on delay & relaxation oscillations 2.5 Simulations Experiments (diode laser) S Time (ns) A linear stability analysis of the rate equations allows to calculate the RO frequency RO 1 p Adapted from J. Ohtsubo 06/01/

24 Variation of the relaxation oscillation frequency with the output power RO 1 S -1 p Laser on: RO S p 06/01/

25 Turn-on transient of a d 3+ :YAG laser ote the time-scale: for diode lasers is a few ns From T. Erneux and P. Glorieux, Laser Dynamics (CUP 2010) 06/01/

26 Phase-space representation S, plane For the d 3+ :YAG laser What is D? D=(dI/dt)/I+1 06/01/

27 Turn-on of a multi-mode laser Parabolic gain profile: Winner takes all: after transient mode-competition, the mode with maximum gain coefficient wins. But non-transient competition has been observed. More advanced gain models allow explaining non-transient competition. From Ohtsubo (Semiconductor lasers) 06/01/

28 Triangular signal: LI curve S Slow quasi-static current ramp (T=200 ns) Time (ns) S Pump parameter, 06/01/

29 With a fast ramp: turn-on delay 1.5 T=20 ns Experiments S Time (ns) 06/01/2016 Tredicce et al, Am. J. Phys., Vol. 72, o. 6, June

30 Dynamical hysteresis Simulations Experiments S Pump 06/01/

31 Influence of gain saturation ds dt 1 sp G S GS 1 p d dt 1 G(, S) 1 S =0 = S Damped relaxation oscillations Pump Pump Gain saturation takes into account phenomenologically several effects (spatial and spectral hole burning, thermal effects) 06/01/

32 Injection current modulation 1.5 Digital 1 2 Analog <S> S Time (ns) Time (ns) Direct modulation allows to encode information in the laser output power ( amplitude modulation ) 06/01/

33 Why modulation is important? Optical waves can be modulated in Amplitude, Phase and in Frequency in order to carry information Adapted from H. Jäckel, ETHZ 06/01/

34 Weak sinusoidal modulation: influence of the modulation frequency = dc + A sin mod t dc = 1.5, A=0.1 For =1.5: RO = 3.56 GHz S T mod = 0.1 ns ns ns (=T RO ) Time (ns) The laser intensity (S = photon density) is modulated at the same frequency of the pump current (), but the phase of the intensity and the current are not necessarily the same. 06/01/

35 Small-signal modulation response Oscillation amplitude diode-pumped d 3+ :YAG laser Analytical expressions can be calculated from the linearization of the rate equations. mod / RO Modulation frequency (KHz) 06/01/

36 Resonance at mod = RO RO 1 p For a semiconductor laser (VCSEL) Adapted from R. Michalzik (VCSELs 2013) 06/01/

37 Large-signal modulation response Experiments Simulations Adapted from J. Ohtsubo 06/01/ ( t) 0 1 msin mod t

38 Hysteresis induced by strong modulation The figures represents the maxima of the oscillations as a function of the normalized frequency near the onset of hysteresis (left), and far away from the onset of hysteresis (right). Hysteresis cycle obtained by slowly changing the modulation frequency forward (full line) and then backward (broken line). The additional smaller jump near ω mod /ω RO =1/2 is a signature of another resonance. The laser is a d 3+ :YAG laser subject to a periodically modulated pump. 06/01/

39 Summary The simple rate equation model for the photon and carrier densities allows to understand the main features of the laser dynamics when the injection current varies: The turn on delay & relaxation oscillations The LI curve (static & dynamic) The modulation response (small and large signal response) 06/01/

40 TF test Semiconductor lasers are described by two rate equations, one for the photon density and another for the carrier density. The laser relaxation oscillation (RO) frequency is proportional to the injected current. The delay in the laser turn-on is independent of the current ramp. The modulation bandwidth has a resonance at the RO frequency. Small and large amplitude modulation result in sinusoidal oscillation of the output power. The relative phase of the output power and input signal depends on the modulation frequency. In a multimode model with a parabolic gain profile, mode competition is a transient dynamics. 06/01/

41 RATE EQUATIO MODEL FOR THE OPTICAL FIELD -ALPHA FACTOR, LIEWIDTH & ITESITY OISE -OPTICAL PERTURBATIOS (IJECTIO, FEEDBACK) -POLARIZATIO ISTABILITIES 06/01/

42 Laser linewidth Schematic representation of the change of magnitude and phase of the lasing field E due to the spontaneous emission of one photon. E y S E 2 E E x ie y E x The line-width of gas and solid-state lasers is well described by the classic Schawlow-Townes formula (f1/p) 06/01/

43 Schawlow-Townes formula ST h c P out 2 c =half-width of the cavity resonance Physical interpretation: In each round-trip, some noise (spontaneous emission) is added to the circulating field It changes the amplitude and the phase of the field. Amplitude fluctuations are damped: the power returns to values close to the steady state. For phase fluctuations, there is no restoring force. Therefore, the phase undergoes a random walk, which leads to phase noise, which causes a finite line-width. But the line-width of semiconductor lasers is significantly higher. 06/01/

44 In diode lasers the enhancement of the line-width is due to the dependence of the refractive index (n) on the carrier density () Sn Henry introduced a phenomenological factor (α) to account for amplitude phase coupling. The linewidth enhancement factor α is a very important parameter of semiconductor lasers. Typically = ST 06/01/

45 Rate equation for the slowly-varying optical field Photon density ds dt 1 1 p S S E sp 2 0 ( t) E( t) e Complex field de dt i t p E E x ie y 1 (1 i)( 1) E 2 sp k x i y 1 sp, D 2 p 0 de dt de dt x y k( k( 1)( E x 1)( E E x E factor y y ) ) D x D y Langevin stochastic term: complex, uncorrelated, Gaussian white noise Derivation of the equations: Ohtsubo Cap. 2 06/01/

46 Rate equation for optical phase de dt de dt x y ds 1 k( 1)( Ex E y ) D 1S D x dt k( E 1)( E E x x ie E y y ) Se D i y d dt d dt p 2 p 1 ( t) GS E y 1 ( t) The instantaneous frequency depends on the carrier density. is a slave variable. The noise sources are not independent. S E x 06/01/

47 These equations allow to understand the optical spectrum of a semiconductor laser In EELs: L= μm = GHz. Because the gain bandwidth is THz longitudinal modes. The line-width of each longitudinal mode depends on the alphafactor and is of the order of 10 MHz. Multimode and single-mode optical spectra (from J. M. Liu, Photonic devices) FFT of E(t) 06/01/

48 These equations also allow to understand the Relative Intensity oise (RI): FFT of S(t) The laser output intensity is detected by a photo-detector, converted to an electric signal and sent to a RF spectrum analyzer. The RI is a measure of the relative noise level to the average dc power. umerical expression Experimentally measured (VCSEL) Peak at the relaxation oscillation frequency RO Adapted from J. Ohtsubo 1 p Adapted from R. Michalzik (VCSELs 2013) 06/01/

49 RATE EQUATIO MODEL FOR THE OPTICAL FIELD -DIODE LASER LIEWIDTH & ITESITY OISE -RESPOSE TO OPTICAL PERTURBATIOS -IJECTIO -FEEDBACK 06/01/

50 Optical perturbations Solitary diode lasers display a stable output (only transient oscillations) but they can be easily perturbed by injected light and can display sustained periodic or irregular oscillations. Optical injection Optical feedback Optical isolator 06/01/2016 Adapted from M. Sciamanna PhD thesis (2004) 50

51 Optical Injection m s Detection system (photo detector, oscilloscope, spectrum analyzer) Two Parameters: o Injection ratio o Frequency detuning = s - m = m Dynamical regimes: o Stable locking (cw output) o Periodic oscillations o Chaos o Beating (no interaction) 06/01/2016 Adapted from J. Ohtsubo 51

52 Model for the injected laser Optical field (t) = E(t) exp (i s t); E(t) = slowly varying amplitude Without injection: With injection: de dt d dt p (1 de dt i)( μ-- E : pump current parameter 1 (1 i)( 1) E 2 p 1) E 2 ie D P inj optical injection from master laser P inj : injection strength = s - m : detuning D Typical parameters: = 3, p = 1 ps, = 1 ns, D=10-4 ns -1 sp0 D ( t) spontaneous emission noise 52

53 Injection locking Experimental verification Beat frequency =0 = s - m Model prediction: the locking range is proportional to the relative injection strength. d 3+ :YAG laser 06/01/

54 Injection locking increases the resonance frequency and the modulation bandwidth 06/01/

55 Outside the injection locking region: regular intensity oscillations The frequency of the intensity oscillations, f 0, can be controlled by tuning the injection strength and the detuning. 06/01/2016 S-C Chan et al, Optics Express 15, (2007) 55

56 Also outside the locking region: ultra-high intensity pulses ( optical rogue waves ) Distribution of pulse amplitudes for different injection currents Time series of the laser intensity C. Bonatto et al, PRL 107, (2011), Optics & Photonics ews February 2012, Research Highlight in ature Photonics DOI: /nphoton /01/

57 Outline Block 1: Low power laser systems Introduction Semiconductor light sources Models Dynamical effects Applications: telecom & datacom, storage, others

58 Brief summary to come back after the winter holiday SCLs have many applications (in dollars, diode lasers are about 50% of laser market). Operating principle: electron-hole stimulated recombination (LEDs: spontaneous recombination) Direct compound semiconductors (GaAs, InGaAs, etc.) are used: the bandgap determines the emitted wavelength. Two main geometries: edge-emitting & VCSELs. SCLs are well described by simple rate equation models. 06/01/

59 Diode laser model (& other class B lasers) ds dt 1 1 p S sp d dt 1 S RO 1 p 06/01/

60 Model for single-mode diode laser: stochastic rate equation for the slowly-varying optical field Sn 0 ( t) E( t) e i t S E 2 E E x ie y E y de dt 1 (1 i)( 1) E 2 p D x i y E x sp D 06/01/

61 Influence of optical perturbations Injection from another laser Injection locking Regular oscillations Chaos & extreme optical pulses Isolator de dt 1 2 p (1 i)( 1) E ie P inj D ( t) Self optical feedback 06/01/

62 L ext Optical feedback induced regimes Tkach and Chraplyvy diagram Regime I: line-width narrowing/ broadening (depending on the phase of feedback), Regime II: mode-hopping, Regime III: single-mode narrowline operation, Regime IV: coherence collapse, Regime V: single-mode operation in an extended cavity mode. 06/01/ Adapted from S. Donati, Laser and Photonics Rev. 2012

63 Feedback induced instabilities negligible small feedback periodic oscillations with weak feedback chaotic oscillations with strong feedback (coherence collapse, Regime IV) I/I th = /01/2016 Adapted from J. Ohtsubo 63

64 Optical feedback effects on the LI curve Coherent feedback Incoherent feedback Feedback-reduced threshold: the amount of reduction quantifies the strength of the feedback. Adapted from R. Ju et al, 06/01/

65 Close to the laser threshold: Low Frequency Fluctuations (LFFs) I/I th = 1.03 I. Fischer et al, PRL

66 LFFs: Complex dynamics, several time-scales L ext 2L ext c Recovery after a dropout: in steps of with relaxation oscillations If L ext = 1 m = 6.7 ns From M. Sciamanna (PhD Thesis 2004) 06/01/

67 Single-mode Lang and Kobayashi Model Optical field (t) = E(t) exp (i 0 t); E(t) = slowly varying amplitude Solitary laser With optical feedback d dt de dt k(1 i)( G 1 2L ext c de dt k(1 i)( 1) E D G E 1) E E( t 2 ) e feedback 0 i D noise = feedback strength = feedback delay time = pump current (control parameters) k 1 2 p sp D 06/01/2016 R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980) 67

68 External cavity modes (ECMs) (t) = E(t) exp (i 0 t) E(t) = E 0 exp [i(- 0 )t] ; (t) = Monocromatic solutions: (t) = E 0 exp (i t) Stable modes (constructive interference) Unstable models (destructive interference) 0 Csin 2 C 1 arctan 1 cos k E 2 0 The number of ECMs increases with: The feedback strength The length of the external cavity 06/01/

69 Laser intensity (arb. units) Good agreement model-experiments With a fast detector: pulses; with a slow detector: dropouts Experiments Stochastic simulations Time (microseconds) Deterministic simulations I. Fischer et al, PRL 1996 A. Torcini et al, Phys. Rev. A 74, (2006) J. Zamora-Munt et al, Phys Rev A 81, (2010) 69

70 Beyond single-mode equations: model for a multi-mode laser Koryukin and Mandel, PRA 70, (2004) Other models have been proposed to explain observations. 06/01/

71 Mode switching with weak optical feedback Furfaro et al, JQE (2005) 06/01/

72 Research at our lab: optical neurons Optical spikes characterise spikes and compare to those of biological neurons 06/01/

73 Extreme feedback-induced optical pulses Observations Simulations Time (in units of ) I. Fischer et al, PRL (2001) J. M. Aparicio Reinoso et al, PRE 87, (2013) 06/01/

74 An example of an application of feedbackinduced chaos: random bit generation After processing the signal, arbitrarily long sequences can be generated at the 12.5-Gbit/s rate. 06/01/2016 Adapted from I. Kanter et al, ature Photonics 2010 & OP 74

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77 Laser synchronization 06/01/

78 POLARIZATIO ISTABILITIES 06/01/

79 EELs: linearly (TE) polarized output Log scale Polarization switching can be induced by polarizationrotated feedback. threshold current 06/01/

80 Intensity (a. u.) In some VCSELs: polarization switching (PS) Circular cavity geometry: two linear orthogonal modes (x, y). Often there is a polarization switching when the pump current is increased. Also hysteresis: the PS points for increasing and for decreasing current are different. The total output power increases/decreases monotonically with pump. 0.5 Drive current X Polarisation Y Polarisation Time (ms) Adapted from Hong and Shore, Bangor University, Wales, UK 06/01/

81 Polarization-resolved LI curve Current-driven PS Stochastic PS Pump current, I/I th -1 Pump current, I/I th -1 06/01/2016 Adapted from J. Kaiser et al, JOSAB 19, 672 (2002) 81

82 Stochastic PS Anti-correlated fluctuations of the two polarizations. Bistability + noise induced switching 06/01/

83 Current-driven PS Type 1: from the Y (low freq) X (high freq) polarization Type 2: from the X (high freq) Y (low freq) polarization Several models have been proposed to explain these PS Adapted from B. Ryvkin et al, J. Opt. Soc. Am. B /01/

84 Thermal shift of the gain curve Y X Birefringence: the polarizations have different optical frequencies When the pump current increases Joule heating different thermal shift of the gain curve and of the cavity modes It explains Y (low freq) X (high freq) Type I PS only Adapted from Y. Hong et al, Elec. Lett. (2000) Adapted from M. Sciamanna PhD Thesis (2004) 84

85 VCSEL spin-flip model Assumes a four-level system in which e/h with spin down (up) recombine to right (left) circularly polarized photons: de dt (1 i)( 1) E ( a i p ) E D d dt dichroism birefringence ( ) j ( ) 2 E 2 E E x y ( E i( E E ) / E ) / 2 2 Carrier recombination Pump: carrier injection spin-flip rate Stimulated recombination Martin Regalado et al, JQE 33, 765 (1997).

86 The SFM model can explain both: Y X and XY PSs a <0 and small birefringence: X Y a >0 and large birefringence: Y X I x, I y Pump current Pump current The model also explains the stochastic PS. From M. S. Torre et al, PRA 74, (2006)

87 The SFM model also predicts Polarization bi-stability Polarization instability PS induced by optical injection Irreversible PS Martin-Regalado et al, JQE 33, 765 (1997) 06/01/

88 Experimental observation of irreversible PS Parallel optical injection Orthogonal optical injection A. A. Qader et al, PTL 25, 1173 (2013) 06/01/

89 Polarization switching can be exploited for generating all-optically regular square-waves Two mutually coupled lasers The switching rate is controlled by the coupling delay time. Also, in EELs, regular square-wave switching can be generated via orthogonal (polarization-rotated) feedback. From C. Masoller et al, PRA 84, (2011) 06/01/

90 2.0 ma 2.6 ma The first example of a free-running diode laser (QD VCSEL) generating chaos. The underlying physics comprises a nonlinear coupling between two elliptically polarized modes. 06/01/

91 Models beyond ordinary differential equations (ODEs) Partial differential equations (pdfs) take into account transverse effects Larsson and Gustavsson, Chapter 4 in VCSELs Editor: Michalzik (2013) 06/01/

92 Transverse effects and polarization When the first-order transverse mode starts lasing, it is, in general, orthogonally polarized to the fundamental transverse mode. 06/01/

93 The models that take into account polarization and transverse effects depend on the transverse optical and current confinement Gain guided Index guided 06/01/

94 Index-guided model In Index-guided VCSELs the mode transverse profile is fixed by the build-in optical cavity Assuming that only two radially symmetric modes can lase: j=0 LP01 j=1 LP11 i=0 x i=1 y One ODE for each mode: de dt j, i k( 1 i)( g 1) E j, i j, i F j, i And the carrier equation has to be modified to include a carrier diffusion term. If the VCSEL is gain-guided, 3D modeling is needed (for details: Chapter 3 of book VCSELs). 06/01/

95 Take home message The dynamics induced by optical perturbations (injection from other laser, optical feedback) can be useful for a number of applications. For appropriated parameters: - Optical injection induces injection locking : the slave laser emits at the same frequency as the master laser. - Optical injection increases the relaxation oscillation frequency (and thus, the laser modulation bandwidth). - Optical injection can induce regular oscillations. - Optical feedback induces single-mode emission and reduces the laser line width. However, both, optical feedback and optical injection can lead to chaotic intensity oscillations with parameter regions where rare and extreme intensity pulses can occur. VCSELs can display a complex interplay of transverse and polarization effects; polarization instabilities can also be exploited for applications (for example, all-optical square waves). 06/01/2016

96 VF Test In semiconductor laser models, the alpha-factor takes into account phenomenologically the change in the refractive index induced by the variation of the carrier density. In the injection locking regime the laser emits its natural wavelength but with larger output power. Strong optical feedback can be used for achieving single-mode emission. The external cavity modes are coexisting monochromatic steadystate solutions, the emitted wavelength depends on the feedback parameters. In VCSELs the polarization switching can be due to thermal effects. In VCSELs a PS always occurs when the pump current is increased, and is accompanied by a change in the transverse optical mode. 07/01/

97 THAK YOU FOR YOUR ATTETIO! Universitat Politecnica de Catalunya