Extreme events (optical rogue waves) in self

Transcription

Extreme events (optical rogue waves)
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 Optics 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
Agostina
Villanueva
Carlos Bonazzola
Mónica Agüero
Noelia
Santos
Axel
Lacapmesure
2
What is a rogue (or freak)
wave?
•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 rogue wave was
reported by a drilling platform in the North Sea in 1995
(Draupner wave).
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.
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 optical fiber
pumped by femtosecond laser pulses (a system which is
usually described by the NLSE).
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).
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-solidstate 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.
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.
8
Warning
Yet, due to time limitations, only the results for the allsolid-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 Modulational Instability condition is crossed.
This result makes this laser attractive as a toy system
to study the formation of ocean RW.
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
needed. Average energy pulse:
0.06 mJ.
Pump diode 40W, 50 A – Forced air or water
cooled. Average energy pulse: 0.24 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.
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).
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.
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.
13
Intensities at two different points
in the laser spot.
dE=6, two Lyapunov >0, X=17mm.
(a-d) Intensity in photodiode 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.
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.
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.
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 predictable than around
↑
First return maps indicate an average pulse.
that ORW are preceded and
followed by pulses of rather
well defined intensity.
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.
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.
19
The End
Please feel free of making further
questions to:
alex.hnilo@gmail.com
mgkovalsky@gmail.com
20
Still on time to apply!
www.eecos.org
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).
22
What follows is auxiliary
material.
23
24
Sketch of the five variables
iterative map
Sn+1 =
Sn
(A+Bρn )2+ (BλSn )2
{ABCD} and {IKJL} are the
Kostenbauder’s round trip
matrices.
(A+Bρn ) (C+Dρn )+ BD (λSn )2
ρn+1 =
(A+Bρn )2+ (BλSn)2
Tn+1 =
Tn
ITn
(K+IQn )2+ ( π )2
= Tn
L-IQn+1
K+IQn
* paraxial beams.
* pulse duration > 10 fs.
* no bandwidth limitation.
* no spatial apertures.
Tn
(K+IQn ) (J+LQn )+ IL( π )2
Qn+1 =
ITn
(K+IQn )2+ ( π )2
µ-1
U* Sn + Un S*
Un+1 = Un {1 – µ (
)
+
4
µ }
Ds
2
S: inverse of beam area
ρ: inverse of beam radius of
curvature
T: inverse of pulse duration,
squared.
Q: pulse chirp
U: pulse energy
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.N0
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.
26
As a further proof:
350
1,0
250
% false nearest neighbors
number of pulses
300
200
150
100
50
real series
surrogated
0,8
0,6
0,4
0,2
0
85
90
95
100
105
110
115
intensity (arbitrary units)
120
125
0,0
2
4
6
8
10
dimension
dE=5, chaotic (one λL>0), kurtosis ≈3, no L-shaped distribution, simple
laser spot, no ORW are observed.
27
As a further proof, #2:
2500
1,0
% false nearest neighbors
Number of pulses
2000
1500
1000
500
50
100
150
200
Intensity (arbitrary units)
250
0,8
0,6
real series
surrogated
0,4
0,2
0,0
2
4
6
8
10
dimension
dE=9, two λL>0, k=9, L-shaped distribution, complex laser spot, ORW are
observed.
28
Interferograms. Transversal
phase coherence.
Intuitively, we
expected:
/
blocked
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.
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
a representative series:
7874 pulses, 50 ns
FWHM, ≈10 KHz.
Number of ORW: 214
Number of pulses
0.16
0.12
0.08
0.04
1000
100
10
0.00
0
10000
20000
30000
0
40000
4
8
12
16
Detector signal amplitude (a.u.)
Typical chaotic oscilloscope trace
31
ORW in the KLM Ti:Sapphire
laser
M4
≈50fs pulses at 80
MHz, λ= 810 nm
≈600 mW average.
Pumped by CW
laser radiation at
532 nm.
P2
P1
MP2
MP1
M1
R
LB
M3
M2
If the prisms are
adjusted
so
that
GVD→0-, the output
becomes chaotic.
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.
P1 mode
(transform-limited
pulses)
Intensidad normalizada
1.0
0.8
0.6
0.4
0.2
0.0
760
780
800
820
840
860
λ (nm)
spectra
distributions
P2 mode (chirped
>0 pulses)
Intensidad normalizada
1.0
0.8
0.6
0.4
0.2
0.0
760
780
λ (nm)
800
820
33
A five variables iterative map
reproduces the results
Magic numbers
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)
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.
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):
from the graph, βORW≈ 10-6 fs-2.
If this is true, then P1 should also show ORW if the
pulse energy is increased.
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.
P2
P1
← Variation of β (curve, left axis) and of
the modulus of the largest eigenvalue
(broken line, right axis) as a function of
the scaled gain Γ/Γ*, for P2 (up) and P1
(down). For P2, β > βORW ≈10-6 fs-2 at
Γ/Γ*<1, while the eigenvalue crosses 1
at Γ/Γ*≈2.5. Instead, for P1 the eigenvalue is >1 at Γ/Γ*≈1.1, but β > βORW at
Γ/Γ*≈1.4.
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 →
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 quasiperiodicities (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 selffocusing 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).
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 quasiperiodicities 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.).
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?
39

Extreme events (optical rogue waves) in self

Transcription

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