Free-carrier driven spatio-temporal dynamics in amplifying silicon waveguides

0.44 MB·

by  Samudra Roy

Free-carrier driven spatio-temporal dynamics in amplifying silicon waveguides
Samudra Roy1,2, Andrea Marini1 and Fabio Biancalana1,3
1Max Planck Institute for the Science of Light, Gu¨nther-Scharowsky-Straße 1, 91058 Erlangen, Germany
2Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur-721302, India and
3School of Engineering & Physical Sciences, Heriot-Watt University, EH14 4AS Edinburgh, United Kingdom
(Dated: January 21, 2014)
Wetheoreticallyinvestigatethefree-carrierinducedspatio-temporaldynamicsofcontinuouswaves
in silicon waveguides embedded in an amplifying medium. Optical propagation is governed by
a cubic Ginzburg-Landau equation coupled with an ordinary differential equation accounting for
the free-carrier dynamics. We find that, owing to free-carrier dispersion, countinuous waves are
4 modulationallyunstableinbothanomalousandnormaldispersionregimesandchaoticallygenerate
1
unstableaccelerating pulses.
0
2
n I. INTRODUCTION posed and practically realized. The paradigm model for
a describing optical propagation in amplifying waveguides
J isrepresentedbythecubicGinzburg-Landau(GL)equa-

0 Silicon photonics is a well established area of research tion, which governs a wide range of dissipative phenom-
2 aimingatexploitingsilicon(Si)asaphotoniccomponent ena [16, 17]. In general, GL systems are rich in nature
for the engineering of integrated optoelectronic devices.
and exhibit some peculiar features, e.g., chaos and pat-
] TheextraordinaryopticalpropertiesofSiopenuppossi-
s tern formation [18, 19]. Conversely to the case of con-
c bilities for novel miniaturized applications, ranging from ventionalKerr solitons arising fromthe balance between
i optical interconnection to bio-sensing [1, 2]. In the mid-
t nonlinearity and dispersion [20], localized stationary so-
p infrared, Si has a high refractive index (n 3.5) and lutions of GL systems - namely dissipative solitons - re-
o ≃
negligible linear extinction. However, in the range 1 µm
. sultfromtheexactcompensationofgainandloss[21]. In
s < λ0 < 2.2 µm, two photon absorption (2PA) is rele- a recentwork,we investigatedthe FC-induced dynamics
c
vant and is responsible for high nonlinear extinction [3].
i ofdissipativesolitonsinSi-basedamplifyingwaveguides,
s Owingtothehighrefractiveindex,lightcanbetightlylo-
demonstrating the self-frequency soliton blueshift [22].
y
calized in subwavelength Si-based waveguides [5], which
h
tremendously enhance nonlinear processes [6], including
p Inthispaper,wetheoreticallyinvestigatethepropaga-
[ 2PA that damps optical propagation and limits the effi-
tion of continuous waves (CWs) in a silicon-on-insulator
1 ciency of Si-based photonic components [7]. As a conse- (SOI) waveguide embedded in erbium-doped amorphous
quenceof2PA,pairsofphotonswithtotalenergygreater
v aluminium oxide (Al2O3:Er+). Owing to the externally
than the Si band-gap (E 1.12 eV) are absorbed and
5 g ≈ pumped active inclusions, small optical waves are expo-
electrons are excited to the conduction band modifying
4 nentiallyamplifiedandinstabilitydevelopsuntilthenon-
9 the Si optical response [4]. In this interesting and rather
linear gain saturation comes into play counterbalancing
4 unexploredoperatingregime,free-carriers(FCs)directly
the linear amplification. Taking full account of FC gen-
. interact with the optical field and introduce novel non-
1 eration and recombination, we calculate the stationary
0 linear effects [8]. A quite similar scenario happens in nonlinear CWs of the system and we investigate their
4 gas-filled hollow core photonic crystal fibers (HCPCFs)
stability. Wefindthat,analogouslytogas-filledHCPCFs
1 in the ionization regime, where accelerating solitons [9]
[10],stationaryCWsareuniversallyunstableinbothnor-
: and universalmodulational instability [10] have been re-
v mal and anomalous dispersion regimes. However,due to
i cently observed. In this case, the free plasma generated
X theinherentnon-conservativenatureofoursystem,mod-
through ionization is responsible for an intense blueshift
ulational instability (MI) does not generate a shower of
r of several hundreds of nanometers of the optical pulse
a solitons asin Ref. [10], but anaccelerating chaotic state.
[11]. Thistremendousdynamicsoccursinamuchsmaller
This scenario ensues from the presence of unstable dis-
scale (blueshift of few nanometers) in Si-based waveg-
sipative solitons, which constitute the strange attractor
uides [8] because of the intimate presence of 2PA, which
of the system [18]. Every sub-pulse generated through
isresponsibleforboththecreationofFCsandfordamp-
MI is accelerated by the FC dispersion and experiences
ing. In principle, losses can be reduced in hybrid slot
self-frequency blue-shift. In turn, the overall dynamics
waveguides [12], but in this case also the extraordinary
accelerates in the temporal domain and blueshifts in the
effects ensuing from FC dynamics are reduced accord-
frequency domain. The paper is organized as follows.
ingly.
In section II we describe the geometry of the system,
An alternative strategy for overcoming losses consists introducing the governing equations and calculating the
of embedding Si waveguides in gaining media. In this nonlinear stationary CW modes. In section III we ana-
context, amplification schemes based on III-V semicon- lytically and numerically study MI of CWs. Analytical
ductors [13], rare-earth-ion-doped dielectric thin films calculationspredictuniversalMI andacceleratingchaos,
[14] and erbium-doped waveguides [15] have been pro- which is confirmed by numerical simulations.
2
II. MODEL
0.2 (a)
We consider a SOI waveguide with lateral dimensions
h = w = 525 nm surrounded by Al2O3:Er+, which gain 0.15
bandwidth is of the order of 100 nm around the car-
rier wavelength λ 1540 nm. In principle, other gain
≃ 2
schemes involving the use of semiconductor active mate- ρ 0.1
rials can be considered and the gain bandwidth can be
increased accordingly [23]. Without any loss of general-
0.05
ity,weassumethattheSOIwaveguideisfabricatedalong
the [¯110] direction and on the [110] [001] surface. In
×
this case, stimulated Raman scattering (SRS) does not
0.5 1 1.5 2
occur for quasi-TM modes [4]. Initially neglecting gain
g
of the external medium, Si nonlinearity and FC genera-
tion,wehavenumericallycalculatedthelinearquasi-TM
0.8
mode e(r⊥), its dispersion β0(ω0) and the effective area (b)
by using COMSOL. At λ0 = 1550 nm, we have then
found that the second order group velocity dispersion 0.6
(GVD) coefficient is β2 2 ps2/m and the effective
area is Aeff 0.145 µm2≃. N−onlinear pulse propagation 0
≃ ρ 0.4
can be modeled within the slowly varying envelope ap- /
ρ
proximation(SVEA)bytakingtheAnsatzfortheelectric
field E(r,t) = Re A(z,t)e(r⊥)eiβ0z−iω0t , where A(z,t) 0.2
is the pulse envelo(cid:2)pe, e(r⊥) is the linea(cid:3)r mode profile,
ω0 = 2πc/λ0 is the carrier angular frequency, c is the
speed of light in vacuum and β0 is the linear propa- 00 0.5 1
gation constant. We approximate the external gaining ε
medium as a two-levelsystem, which is characterizedby c
aLorentzianspectraldistributionofgainaroundthecar-
rier angular frequency ω0: g(∆ω) = g0/(1 + ∆ω2T22), FIG.1: (a)Squaredamplitudeρ2 ofthestationarynonlinear
whereg0 isthedimensionlessgainpeak,T2 40fsisthe CW mode as a function of the normalized gain parameter of
dephasingtime ofAl2O3:Er+ and∆ω =ω ≃ω0 [24]. For theexternalmediumg. (b)Dependenceoftherescaledmode
smalldetuning∆ω <<ω0, thespectralgai−ndistribution amplitudeρ/ρ0 on theFC parameter ǫc for g=1,g2=0.16.
can be approximated by g(∆ω) g0 g2∆ω2T22. FCs
≃ −
affect the optical propagation by means of two mech-
a(tFhneCisomAp)st.:icTaFhlCeps-rieonpdpaurgocaectdeisosdneissispaegrroesviioenrntnre(idnFsCbicyDatl)lhyaenncdoonualpibnlesedoarrepqtauinoadn- [i2s6n]o.rTmhaelizpeadratmoeφtcer=θσcN=cβLTDP,Aw|βh2e|rσeλσ30/≃(116.π435cׯht100n−222)1amc2-
tions cwohuenrteskforFC1A.3,5whi1le0−µ27=m23π[k2c5/,(σ27λ]0.)TachceouchnatsrafocrteFriCstDic,
c
s ≃ ×
i∂ u ∂2u+iαu+(1+iK)u2u (1) FC recombination time (tc =1 ns) is also normalized to
ξ − 2 τ | | the initial pulse width: τc =tc/t0 =1 104.
−i(g0+g2∂τ2)u+(i/2−µ)φcu=0, The stationary CW nonlinear modes×of Eqs. (1,2) are
dφ φ found by setting the Ansatz
c =θ u4 c. (2)
c
dτ | | − τ
c u=ρeiηξ, (3)
Timedurationandpropagationlengtharenormalizedto
φc =φ0, (4)
the initial pulse width t0 = 100 fs and dispersion length
LD =t20/β2 . The envelopeamplitude (A) is normalized whereρ,η arethe mode amplitude andpropagationcon-
| |
t(o4 u1=.5)A/1√0P−01,8mw2h/eWre iPs0th=eKλ0eArreffn/o(n2lπinne2aLrDco)eaffincdienn2to≃f dstφan/td,τan=d 0φ.0 =Insθecrτtciρn4gisthteheAsntsaattioznianrtyonEuqms.be(r1o,2f)FoCnse:
± × c
bulk silicon [25]. In Eq. (1), the parameter s = 1 rep- gets
±
resents the sign of the GVD (s = +1 normal dispersion
and s = 1 anomalous dispersion), while the parame-
ter K = −βTPAλ0/(4πn2) is the 2PA coefficient, where ρ2 = −ǫc+qǫ2c +2ρ20ǫc, (5)
βTPA ≃8×10−12m/Wisthebulk2PAconstant[25]. The η = ρ2+2µK(ρ2−ρ20), (6)
linearlosscoefficient(α 0.2dB/cm)isrenormalizedto
l
thedispersionlength(α=≃αlLD)andcanbeneglectedat where ǫc = K/(θcτc) and ρ20 = g/K. In absence of FCs
λ 1.55µm,where2PAdominates. TheFCdensity(Nc) (ǫc ), the mode amplitude ρ converges to ρ0, which
≃ →∞
3
coincideswiththeCWmodeamplitudeoftheuncoupled
cubic GL equation. Note that the solution above does
not constitute a family, but an isolated fixed point, as it
generally occurs in dissipative systems [16]. In Fig. 1a,
the stationary nonlinear CW mode squared amplitude
(ρ2)isplottedagainstthegainparameteroftheexternal
medium (g). In Fig. 1b, the mode amplitude rescaled
to the GL amplitude (ρ/ρ0) is plotted against the FC
parameter ǫc. Note that ρ ρ0 in the limit where FCs
→
are not excited (ǫ ). The red dot in the figure
c
→ ∞
indicates the exactvalue ofρ/ρ0 forg =1,g2 =0.16and
realistic parameters of Si.
III. MODULATIONAL INSTABILITY
Modulationalinstability(MI)ofthenonlinearCWsta-
tionary mode given in Eqs. (5,6) can be determined
by perturbing it with small time- and space-dependent
waves:
u=(ρ+a)eiηξ, (7)
φc =φ0+b, (8)
wherea,bareperturbationsoftheopticalfieldandofthe
FCdistribution. PluggingtheAnsatzaboveinEqs. (1,2)
andretainingonlylineartermsoneachievesthefollowing
set of coupled equations for a,b:
s
i∂ a=(η+ig)a+ +ig ∂2a+ (9)
ξ 2 τ FIG.2: ContourplotsfortheMIgrowthratehasafunctionof
(cid:16) (cid:17)
normalizedgaing andangularfrequencyΩfor(a)anomalous
i
(1+iK)ρ2(2a+a∗) µ (φ0a+ρb), (s = −1) and (b) normal (s = +1) dispersion regimes. The
− −(cid:18)2 − (cid:19) second order GVD coefficient is |β2| = 2 ps2/m, the gain
db =2θ ρ3(a+a∗) b . (10) dispersion is g2=0.16, and the2PA coefficient is K ≃0.4.
c
dτ − τ
c
We assume that a = a1eϑ+a∗2eϑ∗, b = b0eϑ+b∗0eϑ∗,
where
whereϑ=hξ+iΩτ,histhegrowthrateofthesmallperi-
odicperturbationsandΩistheirangularfrequency. Note A0 = Kρ2+g2Ω2+Lθcρ4, (15)
that φ is real and positive, since it represents the num-
c
1
baberovoeffFoCrsa,gbeninerEaqtesd. (v9i,a102)PwAe. Iancsheiretviengatsheteoefxpalrgeesbsiroanics B0 = (cid:18)4 +g22(cid:19)Ω4+(2g2K+s)ρ2Ω2+
equations for the optical field perturbations a1,a2: +2Lθc(g2 sµ)ρ4Ω2. (16)
−
ih+∗ F +∗TL P +∗TL ∗ a1 =0, (11) If the real part of h1 or h2 is positive, small pertur-
(cid:20) P +T L ih+F +T L (cid:21)(cid:20) a2 (cid:21) bations are amplified during propagation giving rise to
−
instability. In Figs. 2a, 2b, we plot the instability pa-
where P =(1+iK)ρ2, T =2ρ4θc(i/2−µ), and rameterh=max[Reh1,2]asafunction ofthe normalized
gain g and angular frequency Ω for (a) anomalous (AD)
τ−1 iΩ
L= c − , (12) and (b) normal (ND) dispersion. The interplay between
τc−2+Ω2 nonlinearanddispersiveeffectsleadstoinstabilityacross
F =(cid:16)2s +ig(cid:17)Ω2+2P −η−ig+(cid:18)2i −µ(cid:19)φ0(.13) tthuerbcaetniotnraslwfrietqhuaenngcyulabryfcrreeqauteinngciessidfeabllainndgsw. iSthminallthpeesre-
spectral regions are amplified and eventually the nonlin-
Non-trivial solutions can be found by setting the deter- earstationaryCWmodebreaksupintoatrainofpulses.
minant of the coefficient matrix to zero, achieving the Owing to the refractive index change induced by FCs,
instability eigenvalues thesepulses areacceleratedandwe foundthatthe insta-
bility eigenvectors are asymmetrically peaked on a blue-
h1,2 = A0(Ω) A20(Ω) B0(Ω), (14) shifted frequency. Remarkably, FC-induced instability
− ±q −
4
1
(b)
0.8
)
u
a.0.6
(

r
e
w0.4
o
P
0.2
0
−50 −25 0 25 50
∆ λ (nm)
FIG. 4: (a) Accelerating chaotic state arising from the per-
FIG. 3: Contour plot of the maximum instability eigenvalue
turbed stationary CW nonlinear mode in AD. (b) Output
maxΩh as a function of g,g2 in (a) anomalous (s =−1) and
blue-shiftedspectrum(redcurve)andinputspectrum(dashed
(b) normal (s = +1) dispersion regimes. The second order
KGV≃D0c.4o.efficient is |β2|= 2 ps2/m and the 2PA coefficient is betluerescgu=rv1e,)g.2I=n0t.h16e,n|uβm2|e=ric2alpssi2m/mul,atainodnKwe≃us0e.d4.theparam-
can be observed in ND regime, analogously to univer-
sal plasma-induced instability observed in gas-filled hol-
of FCs affects the pulse dynamics also via FCD, which
low core photonic crystal fibers [10]. In Fig. 3 we have
is the sole crucial term responsible for the acceleration
plotted the maximum instability eigenvalue maxΩh as a towardsshortertimes. MI continuously generatesunsta-
functionofg,g2for(a)anomalousand(b)normaldisper- ble dissipative solitons that play the role of the strange
sion. Note that in both regimes, the instability parame-
attractor of the chaotic system, bifurcating, collapsing
terincreaseswithgainandisreducedbygaindispersion.
andcreatingotherpulsesduetotheirinherentinstability.
While in AD generation of FCs counteracts instability,
The characteristic duration of the chaotically generated
inNDitisthesolecrucialtermresponsibleforitsoccur- pulses can be analytically predicted as ∆τ 2π ∆Ω−1,
rence. Notethatalsothevacuumbackgroundisunstable ≃ | |
where ∆Ω is the instability frequency window that can
in supercritical conditions (g0 >0). be calculated directly from Eq. (14). For g = 1 and
Inordertoconfirmourtheoreticalpredictionswehave g2 = 0.16, the analytical prediction for the pulse width
numerically solved Eqs. (1,2) by using split-step fast is ∆τ 1.5 ps, which finds agreement with the numeri-
∼
Fourier transform and Runge Kutta algorithms. Owing cal simulation (see Fig. 4a). The temporal acceleration
to the instabilities of the vacuum and of the stationary ofthe chaoticallygeneratedpulses isaccompaniedinthe
CW modes, noise is amplified and chaotically generates spectraldomainbyablue-shiftofabout10nm,asshown
unstable accelerating pulses (see Fig. 4a). Note that the in Fig. 4b. The main obstacle hampering further blue-
intensity of the pulses is generally higher for short times shiftingisrepresentedbythefiniteamplifyingwindowof
and smaller for longer times. This general behavior fol- the gaining material. Thus, if other gaining media with
lows from the influence of FCA, which is initially zero larger spectral window are used, a larger blue-shift can
at t and grows as time increases. Generation be achieved.
→ −∞
5
IV. CONCLUSIONS modulational instability in both anomalous and normal
dispersion regimes. By numerically solving the govern-
In conclusion, in this work we have investigated an- ing equations we have observed an accelerating chaotic
alytically and numerically the propagation dynamics of dynamics resulting from the inherent instabilities of the
continuous waves in an amplifying silicon-based waveg- system. Ourtheoreticalinvestigationshavebeenfocused
uide. WemodeledopticalpropagationusingaGinzburg- onarealisticallyaccessiblesetup,andourtheoreticalpre-
Landauequationforthe opticalfieldcoupledwitha first dictions can be experimentally verified.
order differential equation accounting for the generation
of free-carriers. We have derived the stationary nonlin-
ear CW mode of the system and we have studied its This researchwas funded by the German Max Planck
stability against small perturbations, finding universal Society for the Advancement of Science (MPG).
[1] B. Jalali, J. LightwaveTechnol. 24, 4600 (2006). tron. 45, 454 (2009).
[2] R.Soref, IEEEJ.Sel.Top.QuantumElectron.12, 1678 [15] L. Agazzi, J. Bradley, M. Dijkstra, F. Ay, G. Roelkens,
(2006). R. Baets, K. W¨orhoff and M. Pollnau, Opt.Express 18,
[3] R.Soref, Nat. Phot. 4, 495 (2010). 27703 (2010).
[4] Q. Lin, O. J. Painter and G.P. Agrawal, Opt. Express [16] N. Akhmediev and A. Ankiewicz, Dissipative Soliton:
15, 16604 (2007). Lecture notes in Physics, (Springer, Birlin, 2005)
[5] P. Cheben, D-X. Xu, S. Janz and A. Densmore, Opt. [17] I.S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99
Express14, 4695 (2006). (2002).
[6] J. Leuthold, C. Koos and W. Freude, Nat. Phot. 4, 535 [18] S. Kishiba, S. Toh and T. Kawahara, Physica D 54, 43
(2010). (1991).
[7] L. Yin and G. P. Agrawal, Opt. Lett.32, 2031 (2007). [19] W.vanSaarloosandP.C.Hohenberg,PhysicaD56,303
[8] C.A. Huskoet al.,Scientific Reports3, 1100 (2013). (1992).
[9] M. F. Saleh et al.,Phys. Rev.Lett. 107 (2011). [20] G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Aca-
[10] M. Saleh, W. Chang, J.C. Travers, P. St. J. Russell and demic Press, California, 2008).
F. Biancalana, Phys.Rev.Lett. 109, 113902 (2012). [21] N. R. Pereira and L. Stenflo, Phys. Fluids 20, 1733
[11] P.H¨olzer,W.Chang,J.C.Travers,A.Nazarkin,J.Nold, (1977).
N.Y.Joly, M.F.Saleh,F.Biancalana and P.St.J.Rus- [22] S. Roy, A. Marini and F. Biancalana, Phys. Rev. A 87,
sell, Phys.Rev.Lett. 107, 203901 (2011). 065803 (2013).
[12] C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bo- [23] C. Gmachl, D. L. Sivco, R. Colombelli, F. Capasso and
gaerts, R. Beats, B. Esembeson, I. Biaggio, T. Michi- A. Y.Cho, Nature415, 883 (2002).
nobu, F. Diederich, W. Freude and J. Leuthold, Nat. [24] G. P.Agrawal, Phys.Rev.A. 44, 7493 (1991).
Phot. 3, 216 (2009). [25] M. Dinu et al.,Appl.Phys. Lett. 82, 2954 (2003).
[13] A. W. Fang, H. Park, Y. Kuo, R. Jones, O. Cohen, D. [26] H. Rong, A. Liu, R. Nicolaescu and M. Paniccia, Appl.
Liang, O. Raday, M. N. Sysak, M. J. Paniccia and J. E. Phys. Lett. 85, 2196 (2004).
Bowers, Mater. Today 10, 28 (2007). [27] Q. Lin et al.,Appl.Phys. Lett. 91, 021111 (2007).
[14] K. W¨orhoff, J.D.B. Bredley, F. Ay, D. Geskus, T.P.
Blauwendraat andM. Pollnau, IEEE J. Quantum Elec-