Discrete vortex solitons and PT symmetry

0.56 MB·English

by  Daniel Leykam

Discrete vortex solitons and PT symmetry
Daniel Leykam1, Vladimir V. Konotop2, and Anton S. Desyatnikov1
1Nonlinear Physics Centre, Research School of Physics and Engineering
The Australian National University, Canberra ACT 0200, Australia
2Centro de F´ısica Te´orica e Computacional and Departamento de F´ısica
Faculdade de Ciˆencias, Universidade de Lisboa, Lisboa 1649-003, Portugal
CompiledDecember 11,2013
We study the effect of lifting the degeneracy of vortex modes with a PT symmetric defect, using discrete
3 vortices in a circular array of nonlinear waveguides as an example. When the defect is introduced, the
1 degenerate linear vortex modes spontaneously break PT symmetry and acquire complex eigenvalues, but
0 nonlinear propagating modes with real propagation constants can still exist. The stability of nonlinear modes
2 depends on both the magnitude and the sign of the vortex charge, thus PT symmetric systems offer new
mechanismstocontroldiscretevortices. (cid:13)c 2013 OpticalSocietyofAmerica
n
OCIS codes: 230.7370,080.6755,190.3270
a
J

6
Nonlinear periodic media allow for efficient localiza- We consider a circular array of N waveguides with

] tion and routing of light signals in the form of discrete Kerr nonlinearity, shown in Fig. 1(a), with gain and
s
c optical solitons [1]. Alongside usual signals encoded in a losslocatedatwaveguides1andN,couplingC between
i soliton’samplitude,thephaseofavortexsolitonalsocar- them,andnearestneighborcouplingnormalizedtounity
t
p riesaquantizedbitofinformation,itstopologicalcharge along the rest of the ring (the difference in the coupling
o
m (TC) [2]. The absolute value of TC, |m|, is the phase constantscanbeintroducedbyvariationofthedistances
.
s winding number, in units of 2π, around the vortex ori- between the respective waveguides),
c
gin,andthesignofTCdeterminesthedirectionofpower
i
s flow, or vortex handedness. Periodic media support sta- i∂zE1+CEN +E2−iγE1+δ|E1|2E1 =0,
y
h ble discrete vortex solitons [3] and allow for robust con- i∂zEn+En−1+En+1+δ|En|2En =0, (1)
p trol overvortex TC by switching between integer values i∂ E +CE +E +iγE +δ|E |2E =0.
z N 1 N−1 N N N
[ m↔−m, known as charge flipping [4–6].

In systems with discrete rotationalsymmetry, suchas Here E (z) (n = 1...N) is a dimensionless electric field
1 n
v a ring of N identical waveguides [6], vortex and anti- in the n−th waveguide, En+N = En, and δ = ±1 is
2 vortex modes with phase ∼ exp(i2πmn/N) are degen- thenonlinearcoefficient.Stationarymodestaketheform
5 erate, here n = 1,2...N is the waveguide number. The E (z) → E exp(iβz), where β is the propagation con-
n n
0
charge-flipping transformation m ↔ −m can be seen stant.Aspectsofthelinear,largeN limitofEq.(1)were
1
as a complex conjugation, or time-reversal operation. It previouslyconsideredinRefs.[12,13].Indiscreteconser-
.
1 is interesting to explore the additional possibilities for vative systems the TC is m = 1 N Arg[E∗E ],
0 TC control which can be offered by dissipative parity- and measures the phase windin2gπ Palno=ng1 the cnonnto+u1r.
3
1 time (PT) symmetric systems [7], which have pure real Jn = 2Im(En∗En+1) is the energy flow from site n to
: spectrabelow the PT-symmetrybreakingpoint.Recent site n+1 [6]. The usual assumption is that nonzero TC
v
studiesofinfinitechainsanddiscreteringsofcoupledop- (m > 0) m < 0 corresponds to the (anti)clockwise cir-
i
X tical waveguides with gain and loss have demonstrated culation of phase and energy around the contour. This
r their direct relevance to applications [8–12]. The exis- definition breaks down for even N at the band edge
a
tence of propagating modes with real spectra in such |m| = N/2, which are multipole modes E ∼ (−1)n
n
arraysreadilysuggeststhe possibility ofthe existenceof without energy circulation (ie. not a vortex).
propagating vortices. Broken T symmetry implies that Theequationforstationarysolutionsinring-typesys-
vorticeswithoppositechargescanbehavedifferently,i.e. tems can be cast in the general form
their degeneracy is lifted.
InthisLetterweshowthatnonlinearpropagatingvor- −βE+HE+δF(|E|2)E =0, H =H0+iγH1 (2)
tex modes can still exist even when the PT symmetry
where E = (E ,...,E )T is a column vector (T stands
breaking threshold is reached for the linear system. We 1 N
for transposition) and the nonlinearity is given by the
exploreinterestingconsequencesofliftedvortexdegener-
diagonal matrix F(|E|2) =diag(|E |2,...,|E |2). Linear
acy. The onset of modulational instability becomes sen- 1 N
operator H is a matrix describing the array without
sitive to both the magnitude and sign of vortex charge, 0
dissipationandH describesthelosses.Oneensuresthat
thusexpandingtheavailabletoolboxforTCcontrol.The 1
thecommutator[H ,P]=0andH P =−PH withthe
sensitivity to the sign of vortex charge is impossible in 0 1 1
“parity” inversion matrix P having only anti-diagonal
systems respecting T symmetry.
nonzero elements, P = δ , and that [H ,T] =
ij i,N+1−j 0,1
1
(a) C 3 (b) 20 (a) 2 0 (b)
E E β 2 P P
1 N
1
E E 0 10 +1 10 +1
2 N-1 -1
-2
-3 0 0
0 0.5 1 1.5 C 2 -1 1 3 β 5 -1 1 3 β 5
3 (c) 3 (d) 20 (c) 2 0 (d)
β β
2 2 P ±1 P
1 1 +1 -1
0 +1 -1 0 10 10 +1
-1 -1
+1
-2 -2 -1
-3 -3 0 0
0 0.5 1 1.5 C 2 0 0.5 1 1.5 C 2 -1 1 3 β 5 -1 1 3 β 5
Fig. 1. (Color online) (a) Schematic of N-site ring with Fig.2.(Coloronline) Stable (unstable) nonlinearmodes
gain (+) and loss (-) at waveguides 1 and N. Phase cir- of N = 3 ring are shown with solid (dashed) lines. TCs
culation direction of vortices with m > 0 and m < 0 areindicatednexttothecurves,m=+1inred(purple),
is indicated by anti-clockwise(red) and clockwise (blue) m = −1 in blue (brown), and m = 0 in black (grey).
arrows.(b)βvs.C foraconservative(γ =0)N =4ring. Parameters are: (a) C = 0.6,γ = 0.2 < γth; (b) C =
Degenerate vortex modes only occur at the intersection 0.6,γ = 0.6 > γth; (c) C = 1.3,γ = 0.2 < γth; and (d)
markedby the black circle. (c) and (d) Linear spectrum C =1.3,γ =0.6>γth.
for N = 3 and N = 4 rings with γ = 0.2. Modes with
TCs +1 (-1) are shown in red (blue).
multipole (lowest β) mode in Fig. 1(d) has m = 1, but
it is not a vortex: there is phase winding, but its phase
does not increase monotonically, resulting in flow from
0,whereT istheoperatorofcomplexconjugation.Thus
the gain site to the dissipative site along both paths. In
[H,PT]=0, i.e. H is PT-symmetric.
contrast the |m| = 1 modes which vanish as C → 1 are
First, we show that if H supports at least one de-
0
true vortices with energy circulation (the sign of J is
generatepair ofvortex modes E belonging to a double n
±
the same for all n).
degenerate eigenvalue β , i.e. H E =β E , and H is
0 0 ± 0 ± 1
Since the nonlinearity in Eq. (2) has the symmetry
symmetric, suchas in Eq. (1), the PT symmetry break-
ing threshold γth, i.e. the value of γ above which the PT (cid:16)F(|E˜|2)E˜(cid:17) = F(|E˜|2)E˜ there exist [11] nonlinear
spectrum has complex eigenvalues, is zero: γ = 0. In- propagating vortex modes bifurcating from each of the
th
deed, from the above properties of H it follows that linear PT-symmetric vortices. These modes do not ex-
0
one can choose E = PE = TE = E∗. Hence haustallpossiblesolitonsolutionswithrealpropagation
± ∓ ∓ ∓
E˜(1) =E +E andE˜(2) =i(E −E )arethetworeal constants which can exist even when PT symmetry is
+ − + −
eigenstates of H and PE˜(j) = (−1)j+1E˜(j) (j = 1,2). broken in the linear regime [11]. In Fig. 2 we present
0
Then, from the Theorem 2.1 of Ref. [14] follows that H families P(β) of nonlinear modes obtained using New-
hasapairofcomplexeigenvaluesforarbitrarilysmall γ. ton’s method (here P = N |E |2 is the total power).
Pn=1 n
The symmetry breakingthresholdofEq.(1) atC 6=1 Note that for γ 6= 0 the total power is not conserved
is non-zero [12]. Hence, there are no vortex eigenstates exceptforstationarymodes.Weconsiderbelowthesim-
of H . Instead, vortex eigenstates appear at nonzero plest N = 3 ring in detail, then discuss properties of
0
γ < γ . Moreover, the PT operator does not change larger N systems.
th
TC, i.e. if E˜ is an eigenmode of H with the charge m, Fig. 2(a) shows the typical N = 3 soliton family for
HE˜ = β˜E˜ at γ < γ , then PTE˜ is also an eigenmode C < 1 with γ < γ . We observe the bifurcation of
th th
withthesameβ˜andm,andthusonecansetPTE˜ =E˜. m = 1 vortices from linear modes at P = 0 discussed
InotherwordsonecanconsiderlinearPT-symmetricso- in Fig. 1(c). As the power is increased, the energy flow
lutions.InFig.1(b)weshowthe spectrumfor anN =4 between E and E in the lower branch decreases and
1 N
ringwhen γ =0.Vortex modes only existatC =1,and eventually changes sign, destroying the vortex, and the
they are degenerate.The effect of nonzero γ is shown in chargebecomeszero.Nom=−1modesarefoundinthe
Figs.1(c,d)foroddN =3withγ =(C2−3C2/3+2)1/3 parameterrangescanned:therearenosuchlinearmodes,
th
and even N =4 with γ =|C−1|. The degeneracy be- andthebrokenT symmetrysuppressestheirsaddle-node
th
comes a pair of exceptional points, and branches with bifurcation.
nonzero topological charge |m| = 1 appear. The break- When γ is increased beyond γ in Fig. 2(b), the lin-
th
ing of T symmetry manifests itself through the separa- ear modes merge and annihilate, and pairs of m = +1
tionofdifferentchargesintodistinctbranches.Curiously, vorticesnowappearatasaddle-nodebifurcation(notice
nonzero m here does not imply nonzero vorticity: the thattheunstablebranchisnotvisibleonthescaleofthe
2
figure). One branch quickly loses its charge and vortic-
ity;this happensatlowerpowersasγ isincreased.Once
again, no m=−1 modes are found.
The situation for C > 1, [Figs. 2(c,d)] is different.
Now nonlinear m = −1 modes exist below γ , because
th
they can bifurcate from linear modes. In this case one
is stable at high power, while the other is unstable, but
both maintaintheir vorticity.A pair of m=+1 vortices
isstillcreatedatasaddlenodebifurcation,withbehavior
similar to Figs. 2(a,b); one branch loses its charge and
vorticity, while the other remains stable. Above the PT
breaking threshold, the m = −1 modes are destroyed,
while the m=+1 saddle node bifurcation remains. Fig.3.(Coloronline)Chargeselectivityinanarraywith
Evidently, the PT symmetric defect in combination N = 6, C = 1, γ = 0.2, and δ = 1. The total power
withtheasymmetriccouplingCdeterminesthechargeof (a)andTC (b) areshownvs.propagationdistancez for
linearvortexmodes-theenergyflowsfromthesitewith m=+2(black,dashed)andm=−2(gray,solid)inputs.
gain to the site with loss along the path with stronger Vortex(red)andantivortex(blue)linesareshownin(c)
coupling, and back via the path with weaker coupling. for input m=+2 and in (d) for m=−2.
In contrast, the charge of nonlinear modes above γ
th
is mainly sensitive to the distribution of gain and loss.
symmetry, the existence, stability and dynamics of non-
These charges are opposite when C >1.
linearvortexmodesbecome sensitivetothe signoftheir
As N is increased modes with larger |m| ≤ N/2 ap-
charge, offering an additional degree of freedom for all-
pear.In conservativesystems with focusing nonlinearity
optical control of discrete vortices.
themodeswithhigher|m|arestable,whilelowercharged
This work was supported by the Australian Research
vorticessufferinstabilitiesaboveacriticalpower[6].This
Council. The work of VVK was supported by the FCT
behavior holds in PT symmetric rings, but the sign of
(Portugal) grants PTDC/FIS/112624/2009 and PEst-
thechargealsoplaysanimportantrole.ForevenN there
OE/FIS/UI0618/2011.
is an additional nonlinear branch bifurcating from the
linear multipole mode, which formally has the highest
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3
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4