Observation of Replica Symmetry Breaking in the 1D Anderson Localization Regime in an Erbium-Doped Random Fiber Laser

0.2 MB·

by  Anderson S. L. Gomes

Observation of Replica Symmetry Breaking in the 1D Anderson Localization Regime
in an Erbium-Doped Random Fiber Laser
Anderson S. L. Gomes,1,∗ Bismarck C. Lima,1 Pablo I. R. Pincheira,1 Andr´e L. Moura,1,2
Mathieu Gagn´e,3 Raman Kashyap,3 Ernesto P. Raposo,4 and Cid B. de Arau´jo1
1Departamento de F´ısica, Universidade Federal de Pernambuco, 50670-901, Recife-PE, Brazil
2Grupo de F´ısica da Mat´eria Condensada, Nu´cleo de Ciˆencias Exatas - NCEx,
Campus Arapiraca, Universidade Federal de Alagoas, 57309-005, Arapiraca-AL, Brazil
3Fabulas Laboratory, Department of Engineering Physics,
Department of Electrical Engineering, Polytechnique Montreal, Montreal, H3C 3A7, Canada
4Laborat´orio de F´ısica Te´orica e Computacional, Departamento de F´ısica,
6
Universidade Federal de Pernambuco, 50670-901, Recife-PE, Brazil
1
(Dated: January 8, 2016)
0
2
The analogue of theparamagnetic tospin-glass phase transition in disordered magnetic systems,

n leading to the phenomenon of replica symmetry breaking, has been recently demonstrated in a
a two-dimensional random laser consisting of an organic-based amorphous solid-state thin film. We
J report here the first demonstration of replica symmetry breaking in a one-dimensional photonic

7 system consisting of an erbium-doped random fiber laser operating in the continuous-wave regime
based on a unique random fiber grating system, which plays the role of the random scatterers and

] operates in the Anderson localization regime. The clear transition from a photonic paramagnetic
s
to a photonic spin glass phase, characterized by the probability distribution function of the Parisi
c
i overlap,wasverifiedandcharacterized. Inthisuniquesystem,theradiationfieldinteractsonlywith
pt the gain medium, and the fiber grating, which provides the disordered feedback mechanism, does
o not interfere with the pump.
.
s PACSnumbers: 42.55.Zz,42.25.Dd,05.40.Fb
c
i
s
y
A Random Fiber Laser (RFL), which is the one- random lasers [15], whereas examples of the latter are
h
dimensional (1D) fiber waveguide version of random ZnO powders [16] or rare earth doped powders, such as
p
[ lasers(RLs)inbulkmaterials,waspioneeredbydeMatos Nd3+ indifferenthosts[17,18]. Althoughoperatingmir-
et al. in 2007[1], usinga colloidalRhodamine 6G+TiO2 rorless, RLs present cavity modes [19, 20]. The optical
1
inthe coreofaphotonic crystalfiber. Itwasfollowedby feedback provided by the scatterers has been classified
v
the demonstration of a RFL based on randomly spaced as incoherent or coherent feedback, although there is no
9
6 fiberBragggratings(FBG)inscribedinanerbium-doped universal agreement. Keeping this notation, when the
3 fiber in 2009 [2, 3]. Soon after, Turitsyn and co-workers incoherent feedback dominates, the spectrum above the
1 demonstrated a RFL in conventional optical fibers via thresholdissmooth,whereasinthecaseofcoherentfeed-
0
Rayleigh scattering due to the refractive index fluctu- back, it presents several narrow spikes on the top of the
.
1 ations, amplified through the Raman effect [4]. While smooth background due to amplified spontaneous emis-
0
the RFL reported in Ref. [1] operated in the so-called sion. These spikes are directly related to the modes of
6 incoherent feedback regime, Hu et al. [5] later demon- the random laser, and were first identified in the work
1
strated the operation of a RFL similar to [1], but with of Ref. [16]. Further work on random lasers is reviewed
:
v
the required properties to operate in the coherent feed- in Ref. [21], including applications and highlights of in-
i
X back regime. Several other reports and applications of terdisciplinary results. Among these, we point out the
r RFL in different types of fibers followed afterwards, as very recent experimental demonstration of replica sym-
a reviewed in Ref. [6]. metry breaking(RSB) inRLs [22], a conceptinherentto
Other 1D or quasi-1D RLs have been reported in the the theory of spin glasses and complex systems [23] and
literature, both from the theoretical and experimental theoretically proposed to be observed in photonic sys-
point of view [7–14]. All of these reports describe the tems, particularly random lasers, in 2006 [24, 25]. In
lasing mechanism or laser related properties of 1D RLs, short, the RSB approach describes how identical sys-
with particular emphasis on the threshold behavior. tems under identical initial conditions can reach differ-
RLs in general differ from conventional lasers as the ent states. By investigating the distribution of correla-
feedbackisprovidedbyscatteringratherthanbyasetof tions between RL intensity fluctuations from pulse-to-
static mirrors. Such scatters can be passive, and there- pulse, the authors of Refs. [24, 25] defined an analogue
fore embedded in a gain medium, or active, playing a to the Parisi overlap order parameter (qmax) and found
double role of scattering and gain medium. Examples of evidence of a transition from the photonic paramagnetic
the former are dye+TiO colloids, which were employed to a glassy phase of light – a photonic spin-glass phase.
2
in the first unambiguous experimental demonstration of The experimental demonstration of Ref. [22] employed
Typeset by REVTEX
2
a RL withfunctionalizedthiophene-basedoligomercom-
monly named (T5OCx) in amorphous solid state in pla- (a)
1.0
nar geometry, pumped by a frequency doubled (532 nm)
y
pulsed (10 Hz, 6 ns) Nd:YAG laser. This 2D RL emit- sit 0.8
n 1.9 mW
tedradiationwiththeseveralspikesdepictingthemodes e
nt
riding on a broad pedestal around 610 nm, when high d i 0.6
raesloowluetriosnpsepctercatrlarlemsoeluastuiornemweanstsewmeprleoyeemdp,laoyseodm. eWwhheant alize 0.4 39 mW
m
smooth spectra was measured. Using the replica the- or
N 0.2
ory, they analyzed the shot-to-shot intensity fluctuation
to obtain the RSB signature and clearly demonstrated
0.0
the photonic paramagnetic to photonic spin-glass phase. 1400 1450 1500 1550 1600 1650
The theoretical and related experimental work has been Wavelength (nm)
reviewed in detail in Refs. [26–28].
(b)
In the present work, we describe the first experimen- 45 15
s)
tal results of RSB in a 1D system, operating in the cw nit
36 12 u
rwdpstwyeorhgsoTmoitivcmehlmihmydeeoet,Eshdopareeepa-msRescscrpaeuFaadrlioLttteaetyaeucbinnrilslnedeeegardsrtpdahlireaynneenrpoetsechabirofitrisbyhsbeeiedreuworvrmfeoienBnrd-sktdcr.aaaowfgtepuategenesddrgitbqerhaRruaescetFkistnLawhgrmaaes(ygteE(iFdmmsru-eBeRavc,GkhiFcew)Lesahr)setaemhtnirnoee-- FWHM (nm) 12879 FWHM (nm)1234000001 Exci2tation p3 ower (4mW) 5000000......012345 Intensity (arbitrary units) 0369 Intensity (arbitrary
0
ployedinthereportofRef.[2],wherethefabricationand
0 17 34 51 68 85
fiber Bragg grating inscription details can be found. For
Excitation power (mW)
the understanding of the present work, it suffices to say
FIG. 1: (color online). (a) Output spectrum of Er-RFL
that a polarization maintaining Er doped fiber was used
at pump powers below (blue curve) and above (red curve)
where a continuous grating with randomly distributed
threshold. (b) FWHM and emitted Er-RFL intensity as a
phase errors was written, instead of a random array of
functionofpumppower. Theinsetshowsacloserlookaround
gratings, as in [3]. This allowed the number of scatter-
thethreshold value. The line is a guide to theeyes.
ers and the randomness to be significantly raised. An
optimized length of 20 cm was used in this work, and
the pump source was a cw laser diode operating at 976 while Fig. 1(b) shows the linewidth reduction and emit-
nm. The threshold and laser linewidth of ≈ 3 mW and tedintensityasafunctionofpumppower. Theinsetgives
0.1 nm (limited by our system’s spectral resolution), re- a closer view around the threshold pump power, where
spectively, were obtained, as compared to ≈ 3 mW and it can be seen that both the emitted intensity and sub-
0.5pmforthesameparametersdemonstratedin[2]. For tle linewidth narrowing occurs at the same pump power
100 mW pump power, a 3 mW output power was mea- around 3mW.
sured [2]. For the results reported here, the maximum ThecharacterizationoftheRSBphasetransitionfrom
pump power employed was 80 mW. As pointed out in thephotonicparamagnetictothespin-glassRLbehavior
Ref. [2], the Er-RFL operates in the Anderson localiza- can be quantified by an overlap parameter qγβ analogue
tion regime, of course in 1D, meaning that the transmis- to the Parisi parameter in spin-glass theory [23]. Two-
sion follows an exponential decay as in the localization point correlations can be calculated either among mode
theory, and studied in detail by Shapira and Fischer for amplitudes aj [25–27] or intensities Ij ∝ |aj|2 [22, 28],
a similar random-grating array system in a single mode though the latter are more accessible experimentally. In
fiber [29]. Therefore, our Er-RFL operates in the reso- particular, by measuring fluctuations in the spectral in-
nant feedback or coherent regime [1, 5, 16, 21]. tensityaveragedoverNs systemreplicas,the overlappa-
For the statistical measurements a sequence of 1000 rameter reads [22, 28]:
spectra was collected for each pump power. To obtain
∆ (k)∆ (k)
the set of 1000 spectra, the output of the Er-RFL was qγβ = Pk γ β , (1)
directed onto an Acton 300i spectrometer with a liquid- ∆2(k) ∆2(k)
qPk γ qPk β
N infrared CCD camera and 0.1 nm resolution at 1530
2
nm. whereγandβ =1,2,...,N ,withN =1000inthiswork,
s s
Figure 1 shows, for the sake of completeness, the Er- denote the replica labels, the average intensity at the
RFL characterization. Figure 1(a) shows the spectrum wavelength indexed by k reads I¯(k) = Ns I (k)/N ,
Pγ=1 γ s
below and above threshold for different pump powers andtheintensityfluctuationisgivenby∆ (k)=I (k)−
γ γ
3
I¯(k). In the present context, with a cw laser as the

pump source, each emission spectrum collected within 10
(a) 1.7 mW
a window time of 100 ms is considered a replica, i.e. a
8
copy of the RL system under fairly identical experimen-
6
tal conditions. In order to confirm that the cw mea-

surementswereappropriate,werepeatedthe experiment 4
and, instead of keeping the laser ON all the time, we
2
employed a chopper before the Er-RFL such as to turn
the pump beam ON and OFF at 200 Hz. The results 0
(b) 3.0 mW
werereadilyreproduced,assuringthatthe statisticalbe- 10
haviorwasmaintained. The probabilitydensity function
(PDF)P(q)describesthedistributionofreplicaoverlaps
q =q , signaling a photonic uncorrelatedparamagnetic 5
γβ
or a RSB spin-glass phase if it peaks at q =0 (no RSB)
3 0
or at values |q|6=0 (RSB), respectively. In this context, x1
thebreakofreplicasymmetrymeansthatthemostprob- P(q) 1800 3.2 mW
able value of the intensity correlations between the laser
(c)
spectra is no longer null, as in the symmetric case, but
120
instead presents a non-zero value.
Figures2(a)to2(d)showtheresultsforthePDFP(q)
60
obtainedfromtheexperimentaldata,andFig.2(e)shows
the value |q| = q at which P(q) has its maximum,
max
0
defining the so-calledParisioverlapparameter. Both re- (d) 82.3 mW
sults are in quite good agreement with the theoretical
6
predictions and the experimental results of Ref. [22]. A
sharptransition coinciding with the threshold, ≈3 mW, 4
is observed from the photonic paramagnetic [Figs. 2(a)
and (b), and excitation power . 3 mW in Fig. 2(e)] to 2
the photonic spin-glass phase with RSB [Figs. 2(c) and
0
(d), and excitation power &3 mW in Fig. 2(e)]. -1 0 1
q
The theoretical background that accounts for the
present findings can be described as follows. In a se-
(e)
riesofremarkableworks[24–28]aphasediagramforRLs 1.0
withdisorderednonlinearmediumhasbeenrecentlybuilt
basedonLangevinequationsforthecomplexamplitudes 0.8 1.0
0.8
of the normalmodes a (t). For open cavity systems, the
general effective Hamiljtonian [27, 28] includes a sum of max0.6 qmax 0.6
q
∗ ∗ ∗ 0.4
quadratic [O(a a )] and quartic [O(a a a a )] disorder 0.4
j k j k l m 0.2
termsinthemodeamplitudes. Thephysicaloriginofthe
0.2 0.0
quadraticcouplingliesinthespatiallyinhomogeneousre- 0 1 2 3 4 5
fractive index, as well as in a nonuniform distribution of 0.0 Excitation power (mW)
the gain and an effective damping contribution due to 0 20 40 60 80
the open cavity leakage. On the other hand, the quartic Excitation power (mW)
coupling is associated to a modulation of the nonlinear
FIG.2: (color online). (a)-(d)PDFof theoverlap parameter
χ(3) susceptibility with a random spatial profile. Such
corresponding to the powers below (a,b), just above (c) and
ingredients are also present in the 1D Er-RFL system
well above (d) the threshold. (e) Dependence of the Parisi
analyzed in this work. overlap orderparameter qmax onthepumppower. Theinset
As the spatial disorder generally makes the explicit details the threshold (solid line is a guide to theeye).
calculationofthe quadraticandquarticcouplingsrather
difficult, they have been considered in Refs. [24–28] as
quenched Gaussian variables. The resulting photonic displaying the presence of photonic paramagnetic, ferro-
Hamiltonian for open-cavity systems thus becomes an magnetic, and RSB spin-glass phases, depending on the
analogue to that of the spherical p-spin model [30], with trend of the disorder to hamper the synchronous oscilla-
a sum of p=2 and p=4 terms and Gaussiancouplings. tion of the modes [24–28].
A replica-trick approach identified a phase diagram for Althoughthe disorderednonlinearmediumconsidered
the pumping rate as a function of the disorder strength, in Refs. [24–28] is 3D, we argue that qualitatively sim-
4
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