Extreme events (optical rogue ) in self-pulsing lasers

Alejandro Hnilo

Centro de Investigaciones en Láseres y Aplicaciones (CEILAP), Instituto de Investigaciones Científicas y Técnicas para la Defensa (CITEDEF), Consejo Nacional de Investigaciones Científicas y tecnológicas (CONICET), Argentina. Non Linear Annual Contractor Review, October 8th, 2015. Basic Research Innovation Collaboration Center (BRICC), Arlington, Virginia.

AFOSR grant FA9550-13-1-0120, “Nonlinear dynamics of self-pulsing all-solid-state lasers”

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 1 Láseres Sólidos Laboratory

Andrés López Marcelo Kovalsky Alejandro Hnilo Mariana Toscani Myriam Nonaka

Carlos Bonazzola Agostina Villanueva Mónica Agüero Noelia Axel Santos Lacapmesure

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 2 What is a rogue (or freak) ?

•The term comes from Ocean science. •It refers to waves of very high amplitude appearing in calm weather. • Rogue waves appear and disappear suddenly. •They do not propagate far. •They are unusual phenomena, but appear much more often than can be expected in a Gaussian distribution. •For a long time, their existence was put in doubt. •The first reliable observation of a was reported by a drilling platform in the North Sea in 1995 (Draupner wave).

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 3 Ocean Rogue Waves

Rogue wave estimated at 60 feet moving away from ship after crashing into it a short time earlier. In the Gulf Stream off Charleston, South Carolina, with light winds of 15 knots. Pictures from NOAA ( 2006)

Rogue waves hit cruise ships.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 4 The causes of rogue waves are not well understood yet

• Some theoretical descriptions of rogue waves assume a Nonlinear Schrödinger Equation (NLSE).

• There are many physical systems described by the NLSE.

• These systems are easier to study experimentally than the waves in the high seas, and may provide hints on the formation of ocean rogue waves.

• In particular, Optical Rogue Waves were observed by Solli et al. in 2007, as fluctuations of the light intensity at the edge of the spectrum produced in a micro-structured pumped by femtosecond laser pulses (a system which is usually described by the NLSE).

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 5 Optical rogue waves, or extreme events. After the pioneering work in 2007, optical rogue waves were observed in many optical systems and devices. Most of them were specially designed to produce large fluctuations of the light intensity.

Our group reported the first observation of optical rogue waves in a standard laser cavity:

(The Ti:Sapphire femtosecond laser was customarily described by a NLSE).

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 6 Why studying optical rogue waves (ORW)?

• In general, optical systems are advantageous “toy systems” to study the phenomenon (easy change of the control parameters and fast record of large sets of statistical data). • In the particular case of the self-Q-switched all-solid- state laser, controlling the formation of ORW would allow the emission of pulses of high intensity at selected times of interest, without having to scale up the whole device. •This is specially interesting for laser rangefinders or target illuminators aboard small unmanned flying vehicles (which is a standard use of these lasers), where the size and weight of power supplies and heatsinks are critical.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 7 When do I get an optical rogue wave? In Ocean dynamics, A fluctuation is considered “rogue” or “extreme” if its amplitude exceeds twice the “Abnormality index” or 4 (sometimes, 8) times the standard deviation. In Optics, additional criteria are: • A distribution with a long tail towards high intensities (“L shaped”). • Kurtosis > 3 (higher tail than a Gaussian). Our Project involves the experimental study of optical rogue waves in self-pulsing lasers (lasers with a nonlinear, or saturable, absorber): • Self (or Kerr-lens)-mode-locked Ti:Sapphire laser (pulses of fs duration at rate 100 MHz); “fast” saturable absorber, λ=808 nm. • Self Q-switched all-solid-state Neodymium laser (pulses of ns duration at rate 10 KHz); “slow” saturable absorber, λ=1064 nm.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 8 Warning

Yet, due to time limitations, only the results for the all- solid-state laser will be described in this talk.

All the main features of the ORW in the Ti:Sapphire have been explained (and recently published: Phys. Rev. A 91, 013836, 2015).

The most important result is that we have established that ORW occur in this laser only if a threshold similar to the condition is crossed. This result makes this laser attractive as a toy system to study the formation of ocean RW.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 9 ORW in the all-solid-state Neodymium laser This laser is made of a diode-pumped active medium and a crystal saturable absorber. The absorber is slow: some “memory” remains of the features of the previous Q-switch pulse. It is not described by the NLSE, but ORW here have a practical interest:

1”

Pump diode 2W, 3 A - No cooling Pump diode 40W, 50 A – Forced air or water needed. Average energy pulse: cooled. Average energy pulse: 0.24 mJ. 0.06 mJ. Echo from a non-cooperative target at 10 Km: OK

Even an increase of a factor 4 above the average pulse (what often occurs during a ORW regime) makes the smaller version useful for a rangefinder or target illuminator.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 10 The Nd:YVO4+Cr:YAG prototype to study ORW The Nd:YVO4 active medium fixes a stable linear polarization operation. The V-shape of the cavity makes the mode diameter to vary strongly near mirror M2. Hence, by adjusting the position (x) of the Cr:YAG crystal, the saturation parameter and the dynamical regime are changed. As x→0, the laser output passes from uniform Q-switching to period doubling cascades, chaos, a period-3 stable window and chaos + ORW. The available theory predicts chaos, but no ORW. We observed ORW only if the Fresnel number is relatively high (#F≈5).

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 11 Two types of observed extreme events: Intensity and time. The two time variables associated with a pulse (ORW or not). In the uniform self-Q-switching regime, Δt+= Δt- = constant. Dimension of embedding (dE) and Lyapunov exponents are calculated from recorded time series, for I, Δt+ and Δt-.

Average intensity = 100

ORW in Intensity are not preceded by an ORW in time, but an ORW in Intensity is followed by an ORW in time.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 12 Time and spatial complexities are correlated

Up: I vs Δt-. Low: color-coded intensity curves of the laser spot. Left: x=7.25mm, dE=7, two positive Lyapunov exponents, complex I vs Δt- diagram, many lobes in the spot. Center: x=7mm, dE=7, period 3, all Lyapunov exponents are negative, period-three window, simple I vs Δt- diagram, few lobes. Right: x=6.75mm, dE=8, one Lyapunov exponent is positive, ORW are observed, complex I vs Δt- diagram, many lobes.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 13 Intensities at two different points in the laser spot.

dE=6, two Lyapunov >0, X=17mm. (a-d) Intensity in B (IBi) vs photodiode A (IAi), for different values of separation (in mm): (a) 0, (b) 1, (c) 3 , (d) 8. Red dots indicate ORW of the simultaneously recorded total intensity time series. Note that ORW in total intensity are not ORW in all the points in the spot!

Histogram of total intensity Kurtosis= 6.02.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 14 Spatial correlation changes with the dynamical regime

Regimes #1 (X=17mm) and #2 (X=5mm) are hyperchaotic with ORW; #3 is periodic. The arrows on the laser spots indicate the initial position and direction the correlation is measured. The horizontal red lines in the series indicate the ORW event threshold. DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 15 Phase coherence is lost in presence of ORW

Contour plots of laser spot and heterodyne interferograms. Upper row: chaotic regime without ORW (dE = 6, one Lyapunov >0, X= 8 mm). Lower row: with ORW (dE= 6, two Lyapunov >0, X= 6 mm). The region marked with a black line indicates where the fringes blur.

Blurring is not observed in absence of ORW even if the regime is chaotic. This suggests the existence of coherence domains and provides further support to the hypothesis that transverse mode interaction is key in the appearance of ORW.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 16 ORW behave more regularly than the average pulse.

→ Superposition of 112 time traces centered at ORW of a time series. The evolution around an ORW is more predict- ↑ able than around First return maps indicate an average pulse. that ORW are preceded and followed by pulses of rather well defined intensity.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 17 Conclusions for the ORW in the all-solid-state laser

• Strong indications that the formation of ORW is related with the nonlinear interaction of several incoherent transversal modes.

•Even though this system is not ruled by the NLSE, its basic mechanism seems close to the intuitive idea on the formation of ocean rogue waves (i.e., mode interaction).

•The trajectory in phase space before and after and ORW seems relatively well defined, this results encourages the goal of predicting and controlling them.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 18 Future work, planned and possible.

•Single-pulse images of the spot (with an ultrafast camera, >104 fps) will determine the features of the patterns associated with an ORW.

• Pumping with a VCSEL (instead that a laser diode) will allow determining the value of #F with precision (and hence, its importance).

•Once a theoretical description is available, devising practical methods to control the formation of ORW in the all-solid-state laser + SA.

•Exploring combinations of all-solid-state lasers with an active medium and SA other than Nd:YVO4 and Cr:YAG will put light on the general causes for the formation of ORW.

•The study of the Ti:Sapphire laser now appears as complete. The study of the chaotic dynamics of femtosecond fiber lasers (also ruled by the NLSE) is a new and almost unexplored field.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 19 The End

Please feel free of making further questions to: [email protected] [email protected]

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 20 Still on time to apply!

www.eecos.org

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 21 Our publications on ORW in lasers

1- “Features of the extreme events observed in the all-solid state laser with a saturable absorber”; C.Bonazzola et al.; arXiv/abs/1506.02014 (2015).

2- “Characteristics of the extreme events observed in the Kerr lens mode locked Ti:Sapphire laser”; A.Hnilo et al.; Phys.Rev.A 91, 013836 (2015).

3- “On the features of the Optical Rogue Waves observed in the Kerr lens mode locked Ti:Sapphire laser”; A.Hnilo et al.; arXiv/abs/1403.5210 (2014).

4- “Optical rogue waves in the solid-state laser with a saturable absorber”; C.Bonazzola et al.; IEEE Xplore paper 3022515 (2013), d.o.i.: 10.1109/LDNP.2013.6777416.

5- “Optical rogue waves in the all-solid-state laser with a saturable absorber: importance of the spatial effects”; C.Bonazzola et al.; J.Opt. 15, 064004 (2013).

6-“Extreme Events in Ultrafast Lasers”; M.Kovalsky et al.; JW4A.33.pdf CLEO Technical Digest © OSA 2012.

7- “Extreme value events in self-pulsing lasers”; A.Hnilo et al.; IEEE Xplore paper 2163377 (2011), d.o.i.: 10.1109/LDNP.2011.6162070.

8- “Extreme events in the Ti:Sapphire laser”; M.Kovalsky et al.; Opt. Lett. 36 p.4449 (2011).

9- “Chaos in the pulse spacing of passive Q-switched all-solid-state lasers”; M.Kovalsky & A.Hnilo; Opt. Lett. 35(20) p. 3498 – 3500, (2010).

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 22 What follows is auxiliary material.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 23 DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 24 Sketch of the five variables iterative map

Sn Sn+1 = 2 2 (A+Bρn ) + (BλSn ) {ABCD} and {IKJL} are the Kostenbauder’s round trip matrices.

2 (A+Bρn ) (C+Dρn )+ BD (λSn ) * paraxial beams. ρn+1 = 2 2 (A+Bρn ) + (BλSn) * pulse duration > 10 fs. Tn L-IQn+1 Tn+1 = = Tn 2 ITn 2 K+IQn * no bandwidth limitation. (K+IQn ) + ( π ) * no spatial apertures.

Tn 2 S: inverse of beam area (K+IQn ) (J+LQn )+ IL( π ) Qn+1 = 2 ITn 2 ρ: inverse of beam radius of (K+IQn ) + ( ) π curvature * * 2 U Sn + Un S µ-1 Un+1 = Un {1 – ( ) + 4 } T: inverse of pulse duration, µ Ds µ squared. Q: pulse chirp U: pulse energy

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 25 The usual approach predicts chaos, but not ORW. This laser system is usually described by the following set of coupled rate equations (D.Y. Tang et al., Opt.Lett. 28, p.325, [2003]):

dI/dt = (2.I /tr)[σg.lg.(N2-N1) – σa.ls.Ns.kA – k] + Ω.(N2-N1).γ21 dN2/dt = P.N0 – (γ20+ γ21)N2 - σg.c.I.(N2-N1) dN1/dt = - γ10.N1 + γ21.N2 + σg.c.I.(N2-N1) dN0/dt = γ20.N2 + γ10.N1 - P.N 0 dNs/dt = γs.(Ns0 – Ns) - σa.c.I.Ns.kA

The ratio kA between the mode area in the active medium and in the SA defines the condition of saturation for Q-switching. It is the main control parameter.

This approach correctly describes many of the features of this laser, including the chaotic dynamics, but it does not predict ORW. Note that this approach assumes a single mode (#F=1) cavity. Bifurcation diagram, control parameter: kA.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 26 As a further proof:

350 1,0 300

250 0,8 real series surrogated 200 0,6 150 0,4 100 number of pulses

50 0,2 % false% nearest neighbors 0 85 90 95 100 105 110 115 120 125 0,0 2 4 6 8 10 intensity (arbitrary units) dimension dE=5, chaotic (one λL>0), kurtosis ≈3, no L-shaped distribution, simple laser spot, no ORW are observed. DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 27 As a further proof, #2:

2500

1,0 2000

0,8 1500 0,6 1000 real series 0,4 surrogated

Number ofNumber pulses 500 0,2 % false% nearest neighbors

0,0 50 100 150 200 250 2 4 6 8 10 Intensity (arbitrary units) dimension dE=9, two λL>0, k=9, L-shaped distribution, complex laser spot, ORW are observed.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 28 Interferograms. Transversal phase coherence.

Intuitively, we expected:

/ blocked

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 29 Summary of the observations in the all-solid-state laser.

•ORW are associated with high (>4) values of dE and complex spatial patterns in the laser spot. •The ORW do not occur in the whole laser spot. •The spatial pattern associated with the ORW is not always the same, even within the same time series. •The time before an ORW is near the average, but the time after an ORW is longer than the average. ORW are simply more efficient (than the average pulse) in depleting the stored energy in the active medium. •Non-ORW regimes are coherent across the whole spot, even if they are chaotic. ORW regimes show domains of coherence compatible with dynamics ruled by the interaction of a few (incoherent) modes. •The time series around an ORW is more regular than around the average pulse.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 30 ORW in the all-solid-state laser: Early results

The first observation (by chance!) of ORW in an all-solid-state laser with saturable absorber (2010). Transverse pumped Nd:YAG + Cr:YAG ≈10 ns, ≈ 10 KHz, 1064 nm, 10 W (#F≈ 100)

experimental results for 0.16

1000 a representative series: 0.12 7874 pulses, 50 ns

0.08 100 FWHM, ≈10 KHz. 0.04 Number of Number pulses Number of ORW: 214 10 0.00 0 10000 20000 30000 40000 0 4 8 12 16 Detector signal amplitude (a.u.) Typical chaotic oscilloscope trace

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 31 ORW in the KLM Ti:Sapphire laser

M4 ≈50fs pulses at 80 P2 MHz, λ= 810 nm P1 ≈600 mW average. MP2 Pumped by CW MP1 laser radiation at 532 nm. M1 R LB M3 M2

If the prisms are adjusted so that GVD→0-, the output becomes chaotic.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 32 ORW are observed in only one of the two chaotic modes

Two attractors, or dynamical modes, are observed to coexist: • P1: transform-limited pulses at the output (route to chaos through quasi-periodicity). • P2: positive-chirped pulses at the output (route to chaos through intermittency). ORW are observed and numerically predicted in the mode P2 only.

1.0 P1 mode 0.8 0.6

(transform-limited 0.4 Intensidad normalizada pulses) 0.2 0.0

760 780 800 820 840 860 λ (nm) distributions spectra 1.0

0.8 P2 mode (chirped 0.6 0.4

>0 pulses) Intensidad normalizada 0.2 0.0

760 780 800 820 λ (nm)

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 33 A five variables iterative map reproduces the results

Chaotic regime of the mode P2. (a) experimental time series, zoom of ≈2000 pulses of 9978 with a total of 205 ORW, kurtosis= 4.91, (b) zoom of the same, (c) theoretical time series obtained from the five-variables iterative map, zoom of ≈2000 iterations of 104 with a total of 226 ORW, kurtosis= 4.98, (d) Magic numbers zoom of the same. Note the intermittent excursions to a regime of pulses of higher energy in both series. Be aware that each point in (b) and (d) is not the sample of a digital oscilloscope, but the energy of a single pulse in the mode-locking train. Average pulse duration: 80 fs.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 34 Self-phase-modulation is larger for P2 than for P1.

This graph suggests that ORW occur if a threshold similar to the modulational instability one is crossed. For the NLSE:

For the Ti:Sapphire laser (from the description by Haus & Silberberg 1986):

-6 -2 from the graph, βORW≈ 10 fs .

If this is true, then P1 should also show ORW if the pulse energy is increased.

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 35 P1 becomes unstable before reaching the ORW threshold

Distributions of pulse energies in the mode P1 as the gain is (numerically) increased. ORW appear for 40% increase (d). Yet, ORW are never observed in the practice in P1.

← Variation of β (curve, left axis) and of P2 the modulus of the largest eigenvalue (broken line, right axis) as a function of the scaled gain Γ/Γ*, for P2 (up) and P1 -6 -2 (down). For P2, β > βORW ≈10 fs at Γ/Γ*<1, while the eigenvalue crosses 1 at Γ/Γ*≈2.5. Instead, for P1 the eigen- value is >1 at Γ/Γ*≈1.1, but β > βORW at Γ/Γ*≈1.4. P1 If the system is forced to start in P1, it rapidly (≈ 2 µs) evolves into P2 leaving no time to ORW to be observed in P1 in the practice →

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 36 Conclusions for the ORW in the KLM Ti:Sapphire laser

• ORW occur in this laser only if the system crosses a threshold similar to the modulational instability one. This explains why ORW are observed in the mode P2, and not in the P1.

• This feature makes the Ti:Sapphire attractive as a “toy system” to study extreme events in other systems ruled by the NLSE, including (possibly) the oceanic rogue waves.

• A specific feature of the ORW in this laser is the existence of quasi- periodicities (the “magic numbers”, not detailed during this talk). They are explained as the residuals of the periodicities of the optical cavity, when subjected to the opposite forces of phase space contraction (due to the presence of transversal apertures) and expansion (due to self- focusing with the appropriate sign).

• The main features of the ORW in this laser have been explained. There only remains how to control the formation of ORW in the practice (difficult task, because of the ≈100 MHz repetition rate).

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 37 General conclusions.

•The KLM (fast SA) Ti:Sapphire laser (ruled by the NLSE) shows ORW only if a threshold condition similar to the modulational instability is crossed. ORW show quasi- periodicities that are residuals of the periodicities of the “cold” optical cavity (not to be expected in oceanic RW).

•The all-solid-state laser + slow SA (not ruled by the NLSE) shows ORW as a consequence of the interaction of several transversal modes. There is hope to predict and control the formation of ORW in this system (practical interest for rangefinders, etc.).

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 38 Summary of all series with measurable dE.

ORW are associated with “complex” spatial patterns What the experiments say about the evolution of the transversal modes?

DISTRIBUTION STATEMENT A – Unclassified, Unlimited Distribution 39