Table Of ContentEngineering optical soliton bistability in colloidal media
Michal Matuszewski
Nonlinear Physics Center, Research School of Physics and Engineering,
Australian National University, Canberra ACT 0200, Australia
We consider a mixture consisting of two species of spherical nanoparticles dispersed in a liquid
medium. We show that with an appropriate choice of refractive indices and particle diameters, it
is possible to observe the phenomenon of optical soliton bistability in two spatial dimensions in a
broad beam power range. Previously, this possibility was ruled out in the case of a single-species
colloid. As a particular example, we consider the system of hydrophilic silica particles and gas
bubblesgenerated in the process of electrolysis in water. The interaction of two soliton beams can
0
lead to switching of the lower branch solitons to the upper branch, and the interaction of solitons
1
from different branches is phase-independentand always repulsive.
0
2
PACSnumbers: 42.65.Tg,42.65.Jx,47.57.-s
n
a
J I. INTRODUCTION considerthesystemofhydrophilicsilicaparticlesandgas
6 bubbles generatedin the process of electrolysis in water.
2
Inanalogywiththeone-dimensionalcase,theinteraction
Spatial opticalsolitons are formedwhen the changeof
of soliton beams can lead to switching to the upper soli-
] nonlinear refractiveindex induces an effective lensing ef-
s tonbranch,and the interactionofsolitons fromdifferent
c fectthatbalancesdiffractionofthelaserbeam[1]. When branches is phase-independent and always repulsive.
i a laser beam passes through a colloidal medium com-
t
p posedofliquidsuspensionofdielectricnanoparticles,the
o optical gradient force acts against particle diffusion, in-
II. MODEL
.
s creasingtherefractiveindexintheregionsofhigherlight
c intensity. The corresponding local change of the refrac-
i We considera mixture consisting oftwospecies ofcol-
s tive index is of the self-focusing type, and allows for cre-
y ationofspatialopticalsolitonsintheformofself-trapped loidalparticlesdispersedinabackgroundliquidmedium.
h We assume that the first (second) group of particles is
beams, as was demonstrated in both theoretical and ex-
p characterized by a refractive index higher (lower) than
perimental studies [2–7]. We note that optical solitons
[ the background medium index. For the sake of clarity,
have been also observed in other soft matter systems,
1 including liquid crystals [8] and polymers [9]. wewillrefertothetwogroupsasto“particles”and“bub-
v bles”, although each of the species can have the form of
1 Recently,itwasshowntheoretically[6]thattheoptical solid state, liquid, or gas, as long as the condition im-
8 responseofacolloidalmediuminthehard-sphereapprox- posed on their refractive indices is fulfilled. In particu-
6 imation allows for the existence of two different stable
lar,bubbles canbe replacedeg.bya solutionofcolloidal
4 soliton solutions for the same beam power, i.e. the soli-
microshells [14] to enhance the system stability. We also
1. ton bistability of the first kind [10, 11]. Moreover,inter- assume purely dielectric response of all the components
0 actionsofthesebistablesolitonshaveinterestingproper- to the laser light and negligible absorption.
0 ties,notfoundinotheropticalsolitonsystems[12]. Soli-
Theopticalgradientforceinducedbyalaserbeamwill
1 ton switching through collisions and phase-independent
have the effect of locally increasing the concentration of
:
v repulsive interactions were demonstrated. The soliton particles anddecreasingthe concentrationofbubbles, as
i bistability, however, has been only predicted in the one-
X was shown previously [4]. If the background concentra-
dimensional case [6], corresponding to a planar waveg-
tion of bubbles (that is, concentration in the absence of
r uide setup or surface waves. In contrast, it was shown
a laser light) is low enough to describe them as a system
thatinthetwo-dimensionalcasethelowersolitonbranch
of noninteracting particles (ideal gas), the same descrip-
becomes unstable [13] due to the intrinsic instability of
tion will hold in the presence of an optical beam. On
multidimensionalsolitonsinKerrmedia. Moreover,even
the other hand, concentration of particles will increase,
in the one dimensional case the power range for which
and interactions between them can become important,
bistable solitons exist is limited.
even if they were negligible at the outset. We take into
Inthispaper,wedemonstratethattheintroductionof account interactions between the particles in the hard
asecondcolloidalspeciesallowsforthebistabilityoftwo- sphere approximation. We also assume that the (other-
dimensionalsolitonbeams. Theideaof“engineering”the wisenegligible)interactionbetweenparticlesandbubbles
nonlinear response of a colloidal medium by mixing sev- is repulsive, which ensures stability of the system.
eral components was first proposedin [4]. We show that The model of nonlinear laser beam propagation in a
withanappropriatechoiceofrefractiveindicesandparti- colloidal suspension in the hard sphere approximation
cle dimensions, it is possible to observethe bistability in waselaboratedin[6]. Inthe recentexperiment[7]itwas
a broadbeam power range. As a particular example, we shown that the optical response of colloidal suspensions
2
of polystyrene beads can be substantially different from whereV =(π/6)d3 isthespherevolumeandδ =(m2
ν ν ν ν−
that predicted by this approximation. We notice, how- 1)/(m2+2). Ifwelookforthesteadystate(−→j =0)inthe
ν ν
ever, that other types of colloidal systems can be rea-
presence of optical field gradient (−→f = (α /4) E 2)
sonably well described by the hard-sphere model [15]. ex ν ∇| |
we obtain
Moreover,it is plausible to expect that the phenomenon
of soliton bistability can be observable also in the case ανβdE 2 d(ρνZν)
ρ | | = , (4)
of “soft” interaction. The necessary requirement here is ν 4 dx dx
the saturation of colloidal particle concentration at high
which can be solved analytically to give the dependence
packingfractions,whichmustoccurdueto limitedavail-
between E 2 and the packing fractions η
able volume. We can therefore treat the hard-sphere ap- | | ν
proximation as the first step to describing and under- β 1 1 η
E 2 = [g(η ) g(η )]= ln b , (5)
standing the physics of more complicated systems. 4| | α p − 0p α η
In our case, the refractive index of colloidal particles p b (cid:18) 0b(cid:19)
np, bubbles nb and the background index nB fulfill the whereg(η)=(3 η)/(1 η)3+lnη, andη0ν is the back-
− −
condition n > n > n . This assumption is neces- ground packing fraction of the ν component.
p B b
sary for observation of the phenomena described below. Assuming relatively low packing fractions, the corre-
We also assume that the particle and bubble diameters sponding nonlinear refractive index change can be ap-
are much smaller than the laser wavelength in the back- proximately calculated using the Maxwell–Garnett for-
groundmedium,dp,db λ0/nB (Rayleighregime). The mula [17]. For low refractiveindex contrast(nν/nB 1)
≪ ≈
osmotic pressure can be calculated from the equation of we have
state [16]
βρΠνν =Zν(ην), ν =p,b (1) n2eff =εeff ≈εB 1+3Xν δνην! . (6)
where the index ν is replaced by p for solid particles Substituting this formula to the Helmholtz equation
sourrbe,fρor ibsutbhbelecso,lβloid=al1p/karBtTic,leΠ(νorisbtuhbebloes)mcootniccenptrreas-- t∇io2nEfo+rkth02ne2esffloEwl=y v0a,rywinegoebntvaeinloptheeofpreolepcatgriactifioenldequu(˜ra)-
tion, Zν (η) is the compressibility, and η = ρ V is defined by E(˜r) = (2/√β)u(˜r)exp(ik0n0z˜), where n0 =
ν ν ν ν
the packing fraction, where Vν denotes the particle vol- nB(1+3 νδνη0ν)1/2
ume. For bubbles we take the ideal gas compressibil-
ity Z (η ) = 1. For solid particles interacting through ∂u 1P i
b b i + 2u+3k2 δ (η η )u + γ u=0,
a hard sphere potential, the Carnahan-Starling formula ∂z˜ 2k0n0 "∇⊥˜ ν ν − 0ν # 2 ν
Z (η ) (1+η +η2 η3)/(1 η )3 gives a very good Xν Xν
p p ≈ p p− p − p (7)
approximationuptothefluid-solidtransitionatη 0.5
p where k = k n = 2πn /λ and the additional last
≈ 0 B B 0
[16]. Thisphenomenologicalformulaisinagreementwith
term on the left hand side accounts for damping due
exact perturbation theory calculations as well as molec-
to Rayleigh scattering from the dielectric spheres. The
ular dynamics simulations. damping coefficients are given by γ =2π5ρ δ2d6/(3λ4)
ν ν ν ν
We assume for the time being that the gradient of the
[18], where λ = λ /n . Additionally, for steady state
concentration of colloidal particles or bubbles ρ(r) is lo- 0 B
solutions, relation (5) gives
cally parallel to xˆ, and consider a small box of volume
dV =dxdS, withlengthdx andnormalsurfacedS. The αp u2 =g(ηp) g(η0p), (8)
| | −
difference in the osmotic pressure exerted on the right α u2 =ln(η /η )
b b 0b
and left surface dΠ gives rise to an effective force acting | |
on the colloidal particles F . It is equal to the external at each point of space.
int
force that is necessary to sustain the concentration gra- We renormalize spatial coordinates according to
dient, and dΠ = F /dS = f ρdV/dS = f ρdx, (x˜,y˜) = (2/3k2)1/2 (x,y) and z˜ = (2n /3kn ) z,
int int int 0 B
− − − × ×
wheref isthe averageforceactingonasingleparticle. obtaining
int
Using Eq. (1) we get d(ρZ)/dx= f ρβ. The particle
− int ∂u 1 i
current density is equal to i + 2u+ δ (η η )u+ Γ η u=0, (9)
⊥ ν ν 0ν ν ν
∂z 2∇ − 2
ν ν
−→j =ρµ(−→f ex+−→f int)=ρµ−→f ex D (ρZ), (2) X X
− ∇
where the renormalized damping coefficients are
where µ is the particle mobility, and D = µ/β is the
diffusion constant. In the ideal gas limit, this equation 4 d 3
becomes Eq. (3) of [4]. Γν = 3π3 1+3 δνη0ν λν δν2. (10)
Let m = n /n be the ratio of the colloidal particle s ν (cid:18) (cid:19)
ν B X
(orbubble) refractiveindex to the backgroundrefractive
FromEq.(9)andformula(10)weconcludethattheeffect
index. Polarizability of a sphere is given by
of scattering losses depends strongly on the ratio of the
α =3V ε n2δ , (3) particle size to the laser wavelength.
ν ν 0 B ν
3
In the following, we consider a particular system
of hydrophilic silica particles and hydrogen or oxygen
nanobubbles generated in the process of electrolysis in
water [19]. Nanobubbles solutions generated in this way
can remain stable for several days. The interaction be-
tween silica particles and bubbles is repulsive indepen-
dently of the distance [20], which ensures the system
stability. We assume that silica particles have diameter
d = 50 nm and n = 1.45 and bubbles have diameter
p p
d =100nmandn =1,whilethewaterrefractiveindex
b b
isn =1.33. Thebackgroundconcentrationsofparticles
B
and bubbles in regionsof low light intensity are takenas
η = 10−3 and η = 10−2, which is consistent with
0p 0b
the experiments [19]. In real systems, it is not possible
to prepare monodisperse solutions. We assume that the
dispersion of sizes is relatively low, which allows us to
use the present model with d and d equal to average
p b
or effective sizes. The relatively long laser wavelength
λ =1064nmallowsforlowerRayleighscatteringlosses.
0
The typical dependence of packing fractions of both
components on the light intensity as well as the total
induced refractive index change ∆n = n n are pre-
eff 0
−
sentedinFig.1. Thebubble packingfractionis depleted
almost completely already at relatively low light inten-
sities due to the large bubble size and the high polariz-
ability. On the other hand, the concentration of silica
particles remains low until the intensity reaches a cer-
tain higher value. Further on, the concentration of the
particles increases exponentially and finally saturates as
the packing fraction becomes larger than 10%. In re-
sult, thanks to the appropriate choice of particle sizes
and concentrations, the refractive index dependence has
a form of two “steps”. The first step is a consequence FIG.1: (Coloronline)(a)Packingfractionsofsilicaparticles
(dashed)andbubbles(dash-dotted),andthetotalinducedre-
of saturation of bubble induced nonlinear index change
fractiveindexchange(solid) vs.thelight intensity. Theinset
following bubble depletion [4], while the second step is
presentsthedependenceinthelowintensitiesrange. (b)The
aneffect ofsaturationofparticleinducednonlinearityat
samepresentedinabilogarithmicscale. Thankstotheappro-
highpackingfractionsduetolimitedavailablevolume[6].
priate choice of particle sizes and concentrations (see text),
Inthenextsectionweshowthatthisartificiallyprepared the refractive index dependence has a form of two “steps”,
nonlinearity allowsfor the bistability of two-dimensional andsupportstwo-dimensionalsoliton bistability. Seetextfor
soliton beams. valuesof parameters.
III. TWO-DIMENSIONAL SOLITONS AND The branches of stable solutions correspond to positive
SOLITON BISTABILITY
slope dP/dκ > 0 while unstable solutions are character-
izedby a negativeslope [10]. The picture showstwo sta-
We now consider two-dimensional spatial soliton solu- ble branches, and within the power range P 4 17 W
≈ −
tions of the attenuation-free version of Eq. (9). We look stable solutions corresponding to both of the branches
forlocalizedsolutionsintheformofcircularlysymmetric exist. These bistable solitons fulfill all the three stabil-
beams u(r)=A(r)exp(iκz)under the condition Γ =0. ity conditions required for their robustness during colli-
ν
The propagationequation (9) reduces to sions [22].
This picture should be compared with the results ob-
1 ∂2 1 ∂
tained in the case of a single colloidal component. The
κA+ + A+ δ (η η )A=0. (11)
− 2 ∂r2 r∂r ν ν− 0ν effect of soliton bistability was predicted in the one-
(cid:18) (cid:19) ν
X dimensional case [6], corresponding to a planar waveg-
The soliton profiles can be obtained using numerical re- uide setup or surface waves, however the range of pow-
laxation methods, see eg. [21]. The dependence of the ers supporting the bistability was significantly smaller.
solitonpowerP = u2drandwidthW =3 ru2dr/P Moreover,it wasshownthatintwo-dimensionalcasethe
| | | |
vs. the propagation constant κ is displayed in Fig. 2(a). lower soliton branch becomes unstable [13] due to the
R R
4
40 20
no particles
a)
30 15
P
)
m )
µ 10 no bubbles b) c) 10W
( (
W P
10 5
W
0 0
-3.5 -3 -2.5 -2 -1.5
log κ
10
10
2) b) 2) c)
W/m 0.1 I 0.η02 W/m I η
2 η 2 5 1 p
1I (100.05 ηp b 0.01 1I (10 ηp 0.5
0 0 0 0
0 5 10 15 0 0.5 1 1.5 2
r (µm) r (µm)
FIG. 3: (Color online) Collision of two identical out of phase
solitons (∆ϕ = π) from the lower branch (left column) and
of two solitons from different branches (right column). The
FIG. 2: (Color online) (a) Soliton power (solid) and width power in each of the beams is equal to P = 15W (left) and
(dash-dotted) vs. the propagation constant κ. Two stable P = 12W (right), and the angle between the beams is 5◦
branches with dP/dκ > 0 exist. The dashed lines depict and 2.5◦, respectively. The consecutive frames correspond to
powerdependenceinthecaseswhenoneofthecolloidalcom- propagation distances z = 0 (top row), z = 165µm (mid-
ponents (particles or bubbles) is removed from the system. dle row), and z = 425µm (bottom row), In the first case,
Bottom panels show the soliton intensity profiles (solid) and repulsive interaction leads to switching of the solitons from
colloidal particle (dashed) and bubble (dash-dotted) packing the lower (a) to the upper branch (c). In the second case,
fractions for two bistable solitons carrying power P = 10W theinteraction isalsorepulsive,independentlyoftherelative
from (b) the lower stable branch and (c) the upper stable phase between the solitons, and leads to destruction of the
branch. In (c), thebubblepacking fraction is negligible. No- wider but weaker soliton.
tice thedifference in thewidth scale.
intrinsic instability of multidimensional solitons in Kerr
media. Theintroductionofthesecondcolloidalspeciesis
thereforenecessaryforthe bistabilityoftwo-dimensional
solitons. Indeed, if one of the components is removed
soliton from the upper branch carrying the same beam
fromthe system,onlyone ofthe stablebranchesremain,
power,thesesolitonscanbeeasilydistinguishedinexper-
see dashed lines in Fig. 2(a). It is clear that the bubbles
iment. We note that in the case ofthe lowerbranchsoli-
determinethepropertiesofthesystemforlowvaluesofκ
tonspackingfractionsofbothcomponentsarebelow1%,
andlowlightintensities,while the particlesarethemain
while the packing fraction of particles is as high as 60%
acting component in the regime of high κ. The bistabil-
in the center of the solitonbeamfromthe upper branch.
ity is the result of combination of the effects that these
This value indicates that the simple Carnahan-Starling
twospecieshaveonthenonlinearresponseofthesystem.
modelthatwehaveusedfordescriptionofparticleinter-
In Figs. 2(b,c) we present examples of bistable soliton actions breaks down, and appearance of ordered dense
profiles corresponding to the soliton power P = 10W. phase canbe anticipated[16]. We note howeverthat the
It is clear that the light intensity of the lower branch breakdown of our model should not lead to qualitative
soliton (b) corresponds to the first “step” from Fig. 1, changes in the beam propagationand soliton properties,
while the upper branch soliton (c) corresponds to the sincethediscrepancyintheparticleconcentration,which
second“step”. Hence,onecancallthe lowerbranchsoli- determines the nonlinear response, is relatively low. On
tons “bubble solitons”, while the upper branch solitons the otherhand, the packingfractionofbubbles is always
are“particlesolitons”,ifreferringtothemainstabilizing below 1% (the backgroundpacking fraction), which con-
component. Sincethewidthofthesolitonfromthelower firmsthatourassumptionoflowbubble concentrationis
branchis approximately7 largerthan the width of the well justified.
×
5
IV. SOLITON INTERACTIONS themodelofeffectivelyincoherentbeamsduetothelarge
difference in propagation constant κ. Despite that each
We proceed to the investigation of interactions of the ofthe solitons appears as anattractive potentialwell for
solitons from the two bistable branches. We consider a the other soliton, repulsive interaction occurs if the col-
collision of two soliton beams angled towards each other lision angle is small enough [12, 23]. To our knowledge,
which initially have the form of two stationary soliton wepresentthe firstexample ofsolitonrepulsionfromat-
solutions, u (x,y) and u (x,y), separated by a distance tractive potential in two dimensions.
1 2
2x large in comparison to their widths. The solitons
0
haveimprintedoppositelinearphasesk ,whichresemble
0
the initial beam tilt, anda constantphase difference ∆ϕ
V. CONCLUSIONS
u(x,y,z =0)=u (x+x ,y)eik0x+u (x x ,y)e−ik0x+i∆ϕ.
1 0 2 0
−
(12)
Wehavederivedthemodelequationsfordescriptionof
The initial profiles are taken as solutions to the un-
asystemoftwocolloidalcomponentsdispersedinaliquid
damped equation (11), and the evolution of the beams
medium in the presence of coherent laser light. We have
is modelled using the full equation (9) with the scatter-
shown that with an appropriate choice of refractive in-
ing losses included.
dicesandparticledimensions,itispossibletoobservethe
InFig.3wepresentresultsforinteractionoftwoiden-
phenomenon of optical soliton bistability in two spatial
tical out of phase solitons (∆ϕ = π) from the lower
dimensions in a broad beam power range. Analogously
branch (a,b,c) and a collision of solitons from different
as in the one-dimensional case, the interaction of soliton
branches (d,e,f). We find that, similarly as in the one-
beamscanleadtoswitchingtothe uppersolitonbranch,
dimensional case [12], interaction of two solitons from
and the interaction of solitons from different branches is
lower branch can lead to switching to the upper branch,
phase-independent and always repulsive. The presented
Fig. 3(c). However, we were not able to observe a sim-
results can have implications for the experiments on op-
ilar phenomenon in the interaction of two solitons from
tical solitons in soft matter media [2–9, 24].
different branches. Instead, the lower branch soliton is
Acknowledgements
destructed in most cases, see Fig. 3(f). On the other
hand,wefindthattheupperbranchsolitonismuchmore
robust and appears in the same form after the collision. This research was supported by the Australian Re-
Nevertheless,wefoundthatthe interactionbetweensoli- search Council and the Research School of Physics and
tonsfromdifferentbranchesisphaseindependentandal- Engineering of the Australian National University. The
waysrepulsive,inanalogywiththe one-dimensionalcase authorwouldliketothankWiesl awKr´olikowskiformany
[12]. This kind of interaction has been explained within valuable discussions.
[1] Y.S.KivsharandG.P.Agrawal,Optical Solitons: From Opt. Lett.33, 2839 (2008).
FiberstoPhotonicCrystals,(AcademicPress,SanDiego, [10] A.E. Kaplan, Phys.Rev.Lett. 55, 1291 (1985).
2003). [11] S.L. Eix and R.H. Enns, Phys.Rev.A 47, 5009 (1993).
[2] A.Ashkin,J.M.Dziedzic,andP.W.Smith,Opt.Lett.7, [12] M. Matuszewski, W. Krolikowski, and Y. S. Kivshar,
276 (1982); P.J. Reece, E. M. Wright,and K.Dholakia, Phys. Rev.A 79, 023814 (2009).
Phys.Rev.Lett.98,203902(2007);C.Conti,G.Ruocco, [13] M. Matuszewski, W. Krolikowski, and Y. S. Kivshar,
and S. Trillo, Phys. Rev.Lett. 95, 183902 (2005). Photonics Lett. Pol. 1, 4 (2009).
[3] R. Gordon, J. T. Blakely, and D. Sinton, Phys. Rev. A [14] P. J. Rodrigo, V. R. Daria and J. Glu¨ckstad, Opt. Ex-
75, 055801 (2007). press 12, 1417 (2004).
[4] R.El-Ganainy, D.N.Christodoulides, C. Rotschild, and [15] G.Bryant,S.R.Williams,L.Qian,I.K.Snook,E.Perez,
M. Segev, Opt.Express 15, 10207 (2007). and F. Pincet, Phys. Rev.E 66, 060501(R) (2002).
[5] R.El-Ganainy,D.N.Christodoulides, Z.H.Musslimani, [16] J.-P.HansenandI.R.McDonald,Theory of SimpleLiq-
C. Rotschild, and M. Segev, Opt.Lett. 32, 3185 (2007). uids, (Elsevier, 3rd ed., 2006).
[6] M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, [17] J. C. M. Garnett, Philos. Trans. R. Soc. London 203,
Opt.Express 16, 1371 (2008). 385 (1904).
[7] W. M. Lee, R. El-Ganainy, D. N. Christodoulides, K. [18] H.C. van de Hulst, Light Scattering by Small Particles
Dholakia, and E. M. Wright, Opt. Express 17, 10277 (DoverPublications, 1981).
(2009); R. El-Ganainy, D. N. Christodoulides, E. M. [19] K.Kikuchi,Y.Tanaka,Y.Saihara,M.Maeda,M.Kawa-
Wright, W. M. Lee, and K. Dholakia, Phys. Rev. A 80, mura, and Z. Ogumi, J. Colloid Interface Sci. 298, 914
053805 (2009). (2006); K. Kikuchi, A. Ioka, T. Oku,Y. Tanaka, Y. Sai-
[8] M. Peccianti, C. Conti, C. Assanto, A. De Luca, and C. hara, and Z. Ogumi, J. Colloid Interface Sci. 329, 306
Umeton,Nature 432, 733 (2004). (2009).
[9] M. Anyfantakis, B. Loppinet, G. Fytas, and S. Pispas, [20] M. Preuss and H.-J.Butt,Langmuir 14, 3164 (1998); S.
6
Assemi, A. V. Nguyen, and J. D. Miller, Int. J. Miner.
Process. 89, 65 (2008).
[21] J.J.Garc´ıa-RipollandV.M.P´erez-Garc´ıa,SIAMJ.Sci.
Comput. 23, 1315 (2001).
[22] R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, Phys.
Rev.A 36, 1270 (1987).
[23] Yu.S. Kivshar, Z. Fei, and L. V´azquez, Phys. Rev. Lett.
67, 1177 (1991); R. H. Goodman, P. J. Holmes, and M.
I. Weinstein, Physica D 192, 215 (2004); C. Lee and J.
Brand, Europhys.Lett. 73, 321 (2006).
[24] C. N. Likos, Soft Matter 2, 478 (2006).
