Nonlinear Bloch modes in two-dimensional photonic lattices

0.31 MB·English

by  Denis Traeger

Nonlinear Bloch modes in
two-dimensional photonic lattices
DenisTra¨ger1,2,RobertFischer1,DragomirN.Neshev1,
6 AndreyA.Sukhorukov1,CorneliaDenz2,WieslawKrolikowski1,
0 andYuriS.Kivshar1
0
1NonlinearPhysicsCentreandLaserPhysicsCentre,CentreforUltrahighbandwidthDevices
2
forOpticalSystems(CUDOS),ResearchSchoolofPhysicalSciencesandEngineering,

n AustralianNationalUniversity,Canberra,ACT0200,Australia
a 2Institutfu¨rAngewandtePhysik,Westfa¨lischeWilhelms-Universita¨t,48149Mu¨nster,Germany
J [email protected]

7
http://www.rsphysse.anu.edu.au/nonlinear-http://www.uni-muenster.de/physik/ap/denz
]
s
c
ti Abstract: Wegenerateexperimentallydifferenttypesoftwo-dimensional
p
Blochwavesofasquarephotoniclatticebyemployingthephaseimprinting
o
technique.WeprobethelocaldispersionoftheBlochmodesinthephotonic
.
s latticebyanalyzingthelineardiffractionofbeamsassociatedwiththehigh-
c
i symmetry points of the Brillouin zone, and also distinguish the regimes
s
of normal, anomalous, and anisotropic diffraction through observations of
y
h nonlinearself-actioneffects.
p
© 2008 OpticalSocietyofAmerica
[

1 OCIScodes: (190.4420)Nonlinearoptics,transverseeffectsin;(190.5940)Self-actioneffects;
v (050.1950)Diffractiongratings.
7
3
0 Referencesandlinks
1 1. J.D.Joannopoulos, R.D.Meade, andJ.N.Winn, PhotonicCrystals: Molding theFlowofLight(Princeton
0 UniversityPress,Princeton,1995).
6 2. P.S.Russell,“Blochwaveanalysisofdispersionandpulse-propagationinpuredistributedfeedbackstructures,”
0 J.Mod.Opt.38,1599–1619(1991).
/ 3. P.St.J.Russell,T.A.Birks,andF.D.LloydLucas,“PhotonicBlochwavesandphotonicbandgaps,”inConfined
s
ElectronsandPhotons,E.BursteinandC.Weisbuch,eds.,(1995), pp.585–633.
c
4. H.S.Eisenberg,Y.Silberberg,R.Morandotti,andJ.S.Aitchison,“Diffractionmanagement,”Phys.Rev.Lett.
i
s 85,1863–1866(2000).
y 5. T.Pertsch,T.Zentgraf,U.Peschel,A.Brauer,andF.Lederer,“Anomalousrefractionanddiffractionindiscrete
h opticalsystems,”Phys.Rev.Lett.88,093901–4(2002).
p 6. M.Loncar,D.Nedeljkovic,T.P.Pearsall,J.Vuckovic,A.Scherer,S.Kuchinsky,andD.C.Allan,“Experimental
: and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl.
v
Phys.Lett.80,1689–1691(2002).
i
X 7. P.E.Barclay,K.Srinivasan,M.Borselli,andO.Painter,“Probingthedispersiveandspatialpropertiesofphotonic
crystalwaveguidesviahighlyefficientcouplingfromfibertapers,”Appl.Phys.Lett.85,4–6(2004).
r 8. S.I.Bozhevolnyi,V.S.Volkov,T.Sondergaard,A.Boltasseva,P.I.Borel,andM.Kristensen,“Near-fieldimaging
a
oflightpropagationinphotoniccrystalwaveguides:explicitroleofBlochharmonics,”Phys.Rev.B66,235204–
9(2002).
9. H.Gersen,T.J.Karle,R.J.P.Engelen,W.Bogaerts,J.P.Korterik,N.F.Hulst,van,T.F.Krauss,andL.Kuipers,
“DirectobservationofBlochharmonicsandnegativephasevelocityinphotoniccrystalwaveguides,”Phys.Rev.
Lett.94,123901–4(2005).
10. R. J.P.Engelen, T.J.Karle, H. Gersen, J. P.Korterik, T.F.Krauss, L.Kuipers, and N.F.Hulst, van, “Lo-
calprobingofBlochmodedispersioninaphotoniccrystalwaveguide,” Opt.Express13,4457–4464(2005),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457.
11. Yu.I.Voloshchenko,Yu.N.Ryzhov,andV.E.Sotin,“Stationarywavesinnon-linear,periodicallymodulated
mediawithhighergroupretardation,”Zh.Tekh.Fiz.51,902–907(1981)(inRussian)[Englishtranslation:Tech.
Phys.26,541–544(1981)].
12. W.ChenandD.L.Mills,“Gapsolitonsandthenonlinearoptical-responseofsuperlattices,”Phys.Rev.Lett.58,
160–163(1987).
13. D.N.ChristodoulidesandR.I.Joseph,“Discreteself-focusinginnonlineararraysofcoupledwave-guides,”Opt.
Lett.13,794–796(1988).
14. C.M.deSterkeandJ.E.Sipe,“Gapsolitons,”inProgressinOptics,E.Wolf,ed.,(North-Holland,Amsterdam,
1994),Vol.XXXIII, pp.203–260.
15. J.Feng,“AlternativeschemeforstudyinggapsolitonsinaninfiniteperiodicKerrmedium,”Opt.Lett.18,1302–
1304(1993).
16. R.F.Nabiev,P.Yeh,andD.Botez,“Spatialgapsolitonsinperiodicnonlinearstructures,”Opt.Lett.18,1612–
1614(1993).
17. S.JohnandN.Akozbek,“Nonlinear-opticalsolitarywavesinaphotonicband-gap,”Phys.Rev.Lett.71,1168–
1171(1993).
18. N.AkozbekandS.John,“Opticalsolitarywavesintwo-andthree-dimensional nonlinear photonicband-gap
structures,”Phys.Rev.E57,2287–2319(1998).
19. S.F.MingaleevandYu.S.Kivshar,“Self-trappingandstablelocalizedmodesinnonlinearphotoniccrystals,”
Phys.Rev.Lett.86,5474–5477(2001).
20. B.J.Eggleton,C.M.deSterke,andR.E.Slusher,“NonlinearpulsepropagationinBragggratings,”J.Opt.Soc.
Am.B14,2980–2993(1997).
21. D.Mandelik,H.S.Eisenberg,Y.Silberberg,R.Morandotti,andJ.S.Aitchison,“Band-gapstructureofwaveg-
uidearraysandexcitationofFloquet-Blochsolitons,”Phys.Rev.Lett.90,053902–4(2003).
22. A.A.Sukhorukov,D.Neshev,W.Krolikowski,andYu.S.Kivshar,“NonlinearBloch-waveinteractionandBragg
scatteringinopticallyinducedlattices,”Phys.Rev.Lett.92,093901–4(2004).
23. C.M.deSterke,“TheoryofmodulationalinstabilityinfiberBragggratings,”J.Opt.Soc.Am.B15,2660–2667
(1998).
24. J.Meier,G.I.Stegeman,D.N.Christodoulides,Y.Silberberg,R.Morandotti,H.Yang,G.Salamo,M.Sorel,and
J.S.Aitchison,“Experimentalobservationofdiscretemodulationalinstability,”Phys.Rev.Lett.92,163902–4
(2004).
25. R. Iwanow, G. I. Stegeman, R. Schiek, Y. Min, and W. Sohler, “Discrete modulational instabil-
ity in periodically poled lithium niobate waveguide arrays,” Opt. Express 13, 7794–7799 (2005),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-7794.
26. C. Wirth, M. Stepic, C. Rueter, and D. Kip, “Experimental observation of modulational instability in self-
defocusing nonlinear waveguide arrays,”InNonlinear GuidedWaves andTheirApplications, Postconference
ed.OSA p.TuB2(OpticalSocietyofAmerica,WashingtonDC,2005).
27. M.Chavet,G.Fu,G.Salamo,J.W.Fleischer,andM.Segev,“ExperimentalObservationofDiscreteModulation
Instabilityin1-DNonlinearWaveguideArrays,”InNonlinearGuidedWavesandTheirApplications,Postcon-
ferenceed.OSA p.WD39(OpticalSocietyofAmerica,WashingtonDC,2005).
28. B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E.Sipe, T. A. Strasser, and R. E. Slusher, “Modulational
instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271
(1998).
29. G.Bartal, O.Cohen, H. Buljan, J. W.Fleischer, O. Manela, and M. Segev, “Brillouin zone spectroscopy of
nonlinearphotoniclattices,”Phys.Rev.Lett.94,163902–4(2005).
30. N.K.Efremidis,S.Sears,D.N.Christodoulides,J.W.Fleischer,andM.Segev,“Discretesolitonsinphotore-
fractiveopticallyinducedphotoniclattices,”Phys.Rev.E66,046602–5(2002).
31. NonlinearPhotonicCrystals,Vol.10ofSpringerSeriesinPhotonics,R.E.SlusherandB.J.Eggleton,eds.,
(Springer-Verlag,Berlin,2003).
32. J.W.Fleischer,M.Segev,N.K.Efremidis,andD.N.Christodoulides,“Observationoftwo-dimensionaldiscrete
solitonsinopticallyinducednonlinearphotoniclattices,”Nature422,147–150(2003).
33. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced
dislocationsinpartiallycoherentphotoniclattices,”Phys.Rev.Lett.92,123902–4(2004).
34. O.Cohen,B.Freedman,J.W.Fleischer,M.Segev,andD.N.Christodoulides,“Grating-mediatedwaveguiding,”
Phys.Rev.Lett.93,103902–4(2004).
35. B.Freedman,O.Cohen,O.Manela,M.Segev,J.W.Fleischer,andD.N.Christodoulides, “Grating-mediated
waveguidingandholographicsolitons,”J.Opt.Soc.Am.B22,1349–1355(2005).
36. R.Fischer,D.Traeger,D.N.Neshev,A.A.Sukhorukov,W.Krolikowski,C.Denz,andYu.S.Kivshar,“Reduced-
symmetrytwo-dimensionalsolitonsinphotoniclattices,”arXivphysics/0509255(2005).
37. O.Manela, O.Cohen,G.Bartal,J.W.Fleischer, andM.Segev,“Two-dimensional higher-bandvortexlattice
solitons,”Opt.Lett.29,2049–2051(2004).
38. G.Bartal,O.Manela,O.Cohen,J.W.Fleischer,andM.Segev,“Observationofsecond-bandvortexsolitonsin
2Dphotoniclattices,”Phys.Rev.Lett.95,053904–4(2005).
1. Introduction
Thestudyofthewavepropagationinopticalperiodicstructuressuchasphotoniccrystals[1]
hasattractedgrowinginterestinrecentyears.Theperiodicphotonicstructuresexhibitunique
propertiesallowingtomanipulatetheflowoflightatthewavelengthscaleandcreatethebasis
for novel types of integrated optical devices. In such periodic dielectric structures, the prop-
agation of light is governed by the familiar Bloch theorem due to the interplay between the
light and the surroundingperiodic structure [2], that introduces the spatially extended linear
waves,theso-calledBlochwaves,astheeigenmodesofthecorrespondingperiodicpotential.
Thus,thepropertiesofelectromagneticwavesinperiodicstructuresarefullydeterminedbythe
Blochwavedispersionwhich,forthespatialbeampropagation,representstherelationbetween
thelongitudinalandtransversecomponentsoftheBlochwavevector.Sinceanyfinitebeamcan
beexpressedasasuperpositionofsuchBlochwaves[3],thebeampropagationinanyperiodic
structureisalso determinedfromthelocaldispersion.Inparticular,thebeampropagationdi-
rectionisdefinedbythenormaltothedispersioncurvewhilethebeamspreadingisgoverned
bythecurvatureofthiscurve.
The study of Bloch waves and their temporal and spatial dispersion provides a key infor-
mationaboutoverallpropertiesofanyperiodicstructure.Inparticular,dependingonthelocal
dispersionandalocalvalueofthewavevector,anopticalbeam(orpulse)experiencenormal,
anomalousor even vanishingdiffraction (or dispersion) [3, 4, 5]. Experimentally,the Bloch-
wave character of electromagnetic waves in photonic crystal waveguides has been deduced
indirectly by detecting the out-of-plane leakage of light [6], by investigating the evanescent
fieldcouplingbetweenataperedopticalfiberandaphotoniccrystalwaveguide[7],andmore
directly by local near-field probing of the intensity distribution in a waveguide [8]. The full
bandstructureofaphotoniccrystalwaveguidehasbeenrecoveredveryrecentlybyemploying
a near-field optical microscope and probing both the local phase and amplitude of the light
propagatingthroughasingle-linedefectwaveguide[9,10].
The Bloch-wavedynamicsin periodic structuresbecomeseven more dramaticin the pres-
ence of the nonlinear medium response that may lead to the formation of strongly localized
structures, discrete and gap solitons [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The properties
oftheBlochmodesofnonlinearperiodicstructureshavebeenextensivelystudiedintheone-
dimensionalgeometries,includingtheBragggratingsandwaveguidearrays[21,22],aswellas
thestudyofmodulationalinstabilityofone-dimensionalwaves[23,24,25,26,27,28].More
recently,theBrillouinzonestructureofnonlineartwo-dimensionalphotoniclatticeswaschar-
acterizedbasedonthe featuresof collectivewavedynamicsforpartiallycoherentmulti-band
excitations[29]. Nevertheless, to the best of our knowledge,no experimentalstudiesof indi-
vidual two-dimensionalBloch waves from differentbands and probingthe Bloch wave local
dispersionhavebeenreportedyet.
The aim of this paper is twofold. First, we probe the local spatial dispersion of the Bloch
modes of a two-dimensionaloptically-inducedphotoniclattice by analyzing the evolution of
linear andnonlinearpropagationmodesassociatedwith the high-symmetrypointsof thefirst
Brillouin zone. In particular, we excite the Bloch waves associated with the high-symmetry
pointsofthetwo-dimensionallatticebymatchingtheiruniquephasestructureandobservedif-
ferentregimesofthelineardiffraction.Second,weemployastrongself-focusingnonlinearity
andstudynonlinearself-actioneffectsforthetwo-dimensionalBlochwaves.Thisallowsusto
probeandcharacterizethespatialdiffractionofeachparticularBlochmode,dependingonthe
curvatureofthedispersionsurfacesatthecorrespondingpointoftheBrillouinzone.
(a) (b) G1
Total internal reflection gap
X1
-2,0
M1
COMPLETE 2D BANDGAP
-2,5
b X2 M2
M
-3,0 Y X
z
y x G
-3,5
G X M G
Fig.1.(a)Experimentalimageofatwo-dimensionaloptically-inducedphotoniclattice,that
isspatiallyperiodicinthetransversedirections(x,y)andstationaryalongthelongitudinal
direction z. (b) Calculated bandgap dispersion b (K). Dots indicate the main symmetry
points.Insetin(b)depictsthecorrespondingfirstBrillouinzone.
2. Two-dimensionalBlochwaves:theoreticalbackground
Westudythepropagationofanextraordinarypolarizedopticalbeam(aprobebeam)inabiased
photorefractivecrystalwithanopticallyinducedtwo-dimensionalphotoniclattice.Weconsider
aspatiallyperiodicpatternoftherefractiveindexintheformofasquarelattice,whichissta-
tionaryinthelongitudinal(z)direction[Fig.1(a)].Thephotoniclatticeisformedbytheinter-
ferenceof fourmutuallycoherentordinarypolarizedopticalbeams. Thisinterferencepattern
[Fig.1(a)],
I (x,y)=I {cos[p (x+y)/d]+cos[p (x−y)/d]}2,
p g
induces a refractive index modulation of the crystal for extraordinarypolarized light via the
strong electro-optic effect [30]. Here x and y are the transverse coordinates, and d is the lat-
tice period. The spatial evolution of the extraordinary polarized beam with a slowly varying
amplitudeE(x,y,z) propagatingalong the lattice is then governedby the followingnonlinear
parabolicequation,
¶ E ¶ 2E ¶ 2E
i +D + +F(x,y,|E|2)E =0, (1)
¶ z (cid:18)¶ x2 ¶ y2(cid:19)
where
g
F(x,y,|E|2)=− (2)
I +I (x,y)+|E|2
b p
describes the refractive index change that includes the two-dimensional lattice itself and the
self-inducedindex change from the probe beam. The parametersused for numericalcalcula-
tions are chosen to match the conditionsof typical experimentsdiscussed below: the dimen-
sionless variablesx, y, z are normalized to the typical scale x =y =1 µm, and z =1mm,
s s s
respectively;thediffractioncoefficientisD=z l /(4p n x2);n =2.35istherefractiveindex
s 0 s 0
ofabulkphotorefractivecrystal,l =532nmisthelaserwavelengthinvacuum,theparameter
I =1 is the constant dark irradiance, g =2.36 is a nonlinear coefficient proportionalto the
b
electro-opticcoefficientandtheappliedDCelectricfield,latticemodulationisI =0.49,and
g
d=23µmisthelatticeperiod.
Fig. 2. Intensity (top) and phase (bottom) of different Bloch modes from the high sym-
metrypointsofthefirstandsecondbandofasquarelattice.Thebluecolorforthephase
distributioncorrespondstothezerophase,whiletheredcolorcorrespondstothep phase.
Such a periodic modulation of the refractive index results in the formation of a bandgap
spectrum for the transverse components of the wave vectors K and K . Then the propaga-
x y
tionoflinearwavesthroughthelatticeisdescribedbythespatiallyextendedtwo-dimensional
eigenmodes, known as the two-dimensionalBloch waves. They can be foundas solutions of
linearizedequation(1)intheform
E(x,y;z)=y (x,y)exp(ib z+iK x+iK y), (3)
x y
wherey (x,y)isaperiodicfunctionwiththeperiodicityoftheunderlyinglattice,andb isthe
propagationconstant.ForasquarelatticeshowninFig.1(a),thedispersionrelationb (K ,K )
x y
is invariantwith respectto the translationsK →K ±2p /d,andthereforeis fullydefined
x,y x,y
byitsvaluesinthefirstBrillouinzone[Fig.1(b,inset)].Thedispersionrelationb (K ,K )for
x y
thislatticeisshowninFig.1(b)wherethehigh-symmetrypointsofthelatticearemarkedby
reddots.
It is important to note that, for the chosen lattice parameters, there exists a full two-
dimensionalbandgapbetweenthefirstandthesecondspectralband.Theexistenceofatypical
bandgapstructureofthelatticewithacompletetwo-dimensionalgapandthehighlynonlinear
properties of the photorefractive crystal make the optically-induced photonic lattice a direct
analogof a two-dimensionalnonlinearphotoniccrystal. Therefore,our experimentsofferan
idealtest-benchforthesimilarphenomenawithhighlynonlinearandtunabletwo-dimensional
photoniccrystalsthatmaybestudiedinthefuturewithfabricatedstructuresinnonlinearmate-
rials.
TheintensityandphasestructureofthecalculatedBlochwavesforthehighsymmetrypoints
of the lattice from the first and second spectral bands are shown in Fig. 2. The upper row
shows the Bloch-wave intensity profiles and the bottom row shows the correspondingphase
structure. As a reference, the first column shows the light intensity of the lattice itself. For
thetwo-dimensionalBlochwavesfromthefirstband,theintensitydistributionofallmodesis
reflectingthestructureofthesquarelattice,withtheintensitymaximacoincidingwiththoseof
the lattice. However,thephase structuredifferssubstantially.Ascan beseen fromFig. 2,the
phaseofthetwo-dimensionalBlochwavesoriginatingfromtheG pointisconstant.Thephase
1
structurebecomesnontrivialforthemodesfromtheX andM points.FortheX (Y )point,
1 1 1 1
the phase represents a stripe-like pattern being constant along one principal direction of the
latticeandexhibitingp phasejumpsalongtheotherdirection.FortheBlochwavesoriginating
fromtheM pointthephasedistributionresemblesachessboardpattern.
1
Ontheotherhand,thetwo-dimensionalBlochmodesfromthesecondspectralbandhavethe
intensitymaximacenteredbetweenthemaximaofthesquarelattice.Thephasestructurehasa
formofstripesorientedalongoneoftheprincipaldirectionsofthetwo-dimensionallatticefor
theX point,orin45◦withrespecttotheprincipalaxesfortheM point.TheG pointappears
2 2 2
to be nearly degeneratewith the propagationconstantsnearly the same for the second, third,
andfourthbands,andwedonotconsiderithere.
Thedifferenceinthephasestructureofthetwo-dimensionalBlochwavestranslatesintothe
differencesin propagationdynamicsforbeamsof afinite size whichspectrumislocalizedin
thevicinityofthecorrespondinghigh-symmetrypointsintheBrillouinzone.Indeed,thealter-
natingphaseisasignatureofstrongBraggscattering,thatmayleadtoanenhanceddiffraction
ofbeamssimilartotheeffectofthedispersionenhancementintheBragggratings[31].There-
fore,thebeamscanexperienceanisotropicdiffractionduetotheasymmetricphasestructureof
thecorrespondingBlochwaves,andthiscanbedetectedbyanalyzingthebeambroadeningin
thelinearregime.
Thesign ofthecurvatureoftherelateddispersionsurfacecan beidentifiedexperimentally
utilizing the nonlinear self-action of the beam. In the case of a medium with positive (self-
focusing)nonlinearity,increasinginputbeamintensitywillresultineitherfocusingordefocus-
ingoftheoutputbeamdependingonwhetherthecurvatureofthedispersionsurfaceisconvex
orconcave,respectively.Acloseexaminationofthespatialdispersiondefinedbythebandgap
spectrumofthelattice [Fig.1(b)]showsthatthebeamsassociatedwith theG andX points
1 2
will experienceself-focusingin both(x,y) directionsdueto theconvexcurvaturealongthex
and y directions. On the other hand, the beams associated with the M point will experience
1
nonlinearself-spreadingduetotheconcavecurvatureatthecorrespondingpointofthedisper-
sionsurface.TotallydifferentbehaviorisexpectedforthebeamsassociatedwiththeX point
1
of the lattice spectrum,asthe curvaturesof the dispersionsurfaceare oppositein the x and y
directions.Suchbeamswillexperienceananisotropicnonlinearresponse:theywillfocusalong
thedirectionoftheconstantphaseandatthesametimewillself-defocusintheorthogonaldi-
rection.ThesymmetrypointM ofthedispersioncurveisdegeneratebetweenthesecondand
2
the third bands,with bothbandshavingopposite butisotropic curvatures.Due to this degen-
eracythenonlinearself-actionofthebeamsassociatedwiththispointwillresultinacomplex
beamdynamics.
3. Experimentalarrangements
Inordertostudyexperimentallythegeneration,formationandpropagationoflinearandnonlin-
earBlochwavesintwo-dimensionalphotoniclattices,weimplementthesetupshownschemati-
callyinFig.3.AnopticalbeamfromacwfrequencydoubledNd:YVO laser,atthewavelength
4
of532nm,issplitbyapolarizingbeamsplitterintotwobeamswithorthogonalpolarizations.
The vertically polarized beam is passed through a diffractive optical element (DOE), which
producestwoorthogonallyorientedpairsofbeams.Anopticaltelescopecombinesthesefour
beamsattheinputfaceofthe photorefractivecrystal,thusforminga two-dimensionalsquare
interferencepatternwhichisstationaryalongthecrystallength(seeinsetinFig.3).Theperiod
of this pattern is 23µm. The crystal is a Cerium doped SBN:60 of 20mm × 5mm × 5mm
biasedexternallywithaDCelectricfieldof4kV/cmappliedalongthec-axis(horizontal).Due
toastronganisotropyoftheelectro-opticeffect,theordinarypolarizedlatticebeamswillprop-
agatelinearlyinsidethecrystal,whileinthesametimeinducingarefractiveindexmodulation
fortheextraordinarypolarized(probe)beam[30].
Thesecond,extraordinarypolarizedlaser beamisexpandedbyatelescopeandilluminates
theactiveareaofaHamamatsuprogrammablephasemodulator(PPM).Themodulatedbeam
isthenimagedbyalargenumericalaperturetelescopeattheinputfaceofthephotorefractive
crystal. A spatial Fourier filter (FF) is placed in the focal plane of the telescope to eliminate
cw laser PPM
532 nm
l/2
PBS
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)
DOE
FF
FF HV
SBN:Ce CCD
Fig.3.Experimentalsetupfortheexcitationoftwo-dimensionalBlochmodes:HV:High
voltage,CCD:camera,FF:Fourierfiltermask,l /2:halfwaveplate;PPM:Programmable
phasemodulator,DOE:Diffractiveopticalelementtoproducefourcoherentbeams,PBS:
Polarizingbeamsplitter.Leftinset:Geometryofthetwo-dimensionalopticallattice.Right
inset:Exampleofaphaseandamplitudeengineeredwaveintheopticallattice.
higher-orderspectralcomponentsandensurethattheopticalbeamenteringthecrystalwillhave
the phaseandamplitudestructurerequiredtomatchthespecific Blochmode.Themodulated
probebeamis combinedwith the lattice ontoa beamsplitter. Thusitwill propagateontothe
inducedperiodicindexmodulationandsimultaneouslywillexperienceastrongnonlinearself-
action at sub-micro-Watt range, due to the strong photorefractivenonlinearity. Both faces of
thecrystalcanbeimagedonaCCDcamerabyahighnumericalaperturelenstocapturebeam
intensitydistribution.
InordertoexciteselectivelydifferentBlochmodesofthetwo-dimensionallattice,theoptical
beammustmatchtheirtransverseamplitudestructure.ThisisachievedbytheuseofPPMthat
convertstheinitiallyGaussianprobebeamintothedesiredamplitudeandphasemodulationat
the frontface of the photorefractivecrystal. For low inputintensity the incidentprobebeam,
representinglinearBlochwave,doesnotaffecttherefractiveindexofthelatticeandhenceits
propagationiscompletelydeterminedbythedispersionattheparticularpointoftheBrillouin
zone.Afinitebeamwilldiffractwitharatedependingonthevalueofthediffractioncoefficients
along the principal directions of the lattice. These diffraction coefficients are determined by
thecurvatureofthedispersionsurfacesalongthexand/orydirections.Withincreasingpower,
nonlinearself-actionofthebeamwillcounteractitsdiffractioninthecaseofnormaldiffraction,
butitwillenhancethebeamspreadinginthecaseoftheanomalousdiffraction.Thesefeatures
ofthenonlinearself-actionoffinitebeamsallowsustoidentifythecharacterofthedispersion
curveswhenthebeamisassociatedwithaspecificBlochmodeofthelattice.
Our experiments are complemented by the numerical simulations of the underlying equa-
tion(1)withtheinitialconditionsmatchingthetransversestructureofthecorrespondingBloch
wavesuperimposedonaGaussiancarrierbeam
E(x,y)=Aexp(x2/w2+y2/w2), (4)
x y
where A is a constant amplitude, w and w are the corresponding beam widths along the x
x y
andyaxes,respectively.Ournumericalsimulationsallowustotrace,withahighaccuracy,the
actualbeamevolutioninsidethecrystalthatisnotdirectlyaccessibleinexperiment,aswellas
providetheopportunitytotestthebeamevolutionforlargerpropagationdistancesbeyondthe
experimentallyaccessiblecrystallengths.
Fig.4.Experimentaldata(toprow)andnumericalresults(bottom)fortheexcitationofthe
two-dimensional Bloch waves from the G point of the first spectral band (G ). (a) Input
1
beam;(b-e)outputsforinputpowersof25nW,125nW,250nW,and375nW,respectively.
Fig.5.Experimentaldata(toprow)andnumericalresults(bottom)fortheexcitationofthe
two-dimensional Blochwaves fromtheX point of thefirstspectral band (X ). (a) Input
1
beam;(b,c)outputsforinputpowersof25nWand375nW,respectively.
4. ExcitationoftheBlochmodesofthefirstband
First,westudyexperimentallythepropagationofbeamsassociatedwithdifferentBlochwaves
fromthefirstspectralbandofthelatticebandgapspectrum(Fig.1).
4.1. G -point
1
TheexcitationofthepointG isrealizedsimplybylaunchingaGaussianbeam[Eq.(4)]along
1
thelatticeandhavingzerotransversewavevectorcomponents.ThestructureofthisBlochwave
is fullysymmetricalongthe principalaxesofthe lattice (Fig. 2).Ifthe initialbeamexcitesa
single lattice site, then the diffractionoutputrepresentsa typical discrete diffraction[32, 33]
and it is well suited to characterizing the induced periodic potential. When the intensity of
the initial beam is high enough,the nonlinearityinducedindexchange leads to a shift of the
propagation constant inside the total internal reflection gap [Fig. 1(b)] and gives rise to the
formationofdiscretelatticesolitons[32,33].
Ourexperimentalresultswereperformedwithaninputbeamofwidthw =w =18µm.For
x y
lowinputpowersof25nW[seeFig.4(a)]thebeamundergoesstrongdiscretediffractiononthe
lattice, where most of its energyis transferredto the outside lobes. With increasing the laser
power[Fig.4(c-e)]thebeamself-focusesleadingtotheformationofadiscretelatticesoliton
inagreementwithpreviousexperimentalstudies[32,33].Thenumericalsimulationsshownin
Fig.4(bottomrow)areingoodagreementwiththeexperimentalobservations.
4.2. X -point
1
The Bloch wave at the X symmetry point of the first band has a strongly asymmetric phase
structure. Thisleads to anisotropicdiffractionfor the propagatingbeamsassociated with this
Fig.6.Experimentaldata(toprow)andnumericalresults(bottom)fortheexcitationofthe
two-dimensional BlochwavesfromtheMpointofthefirstspectralband(M ).(a)Input
1
beam;(b-e)outputsforinputpowersof40nW,125nW,300nW,and850nW,respectively.
Blochmode,allowingfornewtypesofwaveguiding[34,35]duetodifferentcurvaturesofthe
dispersion surface in x and y directions.In orderto balance the rate of beam broadeningdue
todiffractionalongthesedirections,inexperimenttheinputbeamismadeelliptical,elongated
alongthex axis.Itsphaseisconstantalongthey directionandalternatesbyp alongthe per-
pendicularxdirection[Fig.5(a)].Thisstripe-structureislaunchedonsite,i.e.withpositionof
theintensitymaximaonlatticesites.Innumerics,theinputprofileismodeledbythefollowing
expression
E(x,y)=Acos(Kx)exp(x2/w2+y2/w2),
x y
wherew =100µm,w =33µmandK=p /disthelatticewavevector.
x y
Our experimental results and the correspondingnumerical simulations show the same be-
havior for the nonlinear response of the beam at the output face of the crystal [Fig. 5(b,c)
top and bottom row respectively]. At low laser powers, the initial beam spreads strongly in
x-directionduetothelargercurvatureofthedispersionsurface.Increasingbeampowerleads
to strong focusing of the beam along y direction and beam spreading along x axis. This dif-
ferenceinthenonlinearself-actionofthebeamallowsonetoidentifyexperimentallythatthe
dispersionsurfacehasoppositecurvaturesintwo principaldirectionsofthelattice asfollows
fromthetheoreticallycalculatedband-gapdiagram[Fig.1(b)].Theprocessofstrongfocusing
alongthenon-modulatedydirectioniscloselyrelatedtheeffectofgratingmediatedwaveguid-
ing[34,35].
4.3. M -point
1
ThestructureofthedispersionsurfaceneartheMsymmetrypointofthefirstbandissymmetric
inxandydirections.TomatchtheBloch-waveprofile,theinputbeamismodulatedsuchthat
itrepresentshumpsofalternatingphaseintheform
E(x,y)=Acos(Kx)cos(Ky)exp(x2/w2+y2/w2),
x y
withw =w =51µm[Fig.6(a)].
x y
The curvatureof the dispersion surface is concave as indicated in Fig. 1(b). Thereforethe
initialbeamisexpectedtoexhibitenhanceddefocusingwithincreasingofthebeampower.Our
experimentalmeasurementsoftheoutputbeamintensitydistributionaredepictedinFig.6.At
lowlaserpowers(P=40nW)thebeamdiffractslinearlyformingaBlochstate fromtheM -
1
symmetrypoint,showninFig.6(b).Withincreasingpower[Fig.6(c-e)atpowerlevels125nW,
300nW, and 850nW, respectively] the beam defocuses as expected and forms a square type
pattern [Fig. 6(e)]. Similar behavioris also observedin the performednumericalsimulations
[Fig.6,bottomrow].
Fig.7.Experimentaldata(toprow)andnumericalresults(bottom)fortheexcitationofthe
two-dimensionalBlochwavesfromtheXpointofthesecondspectralband(X ).(a)Input
2
beam;(b-e)outputsforinputpowersof20nW,50nW,100nW,and200nW,respectively.
5. ExcitationoftheBlochmodesofthesecondband
The second band of the lattice bandgap spectrum is separated from the first band by a two-
dimensionalphotonicgap.TheBloch modesfromthe topofthe secondband(astheX sym-
metry point) then can be moved by nonlinearity inside the gap, leading to the formation of
spatiallylocalizedgapsolitons.Ontheotherhand,thesecondbandoverlapswiththehigher-
orderbandsattheG andMpointsleadingtodegeneracyofthetwo-dimensionalBlochmodes
andsubsequentlycomplexbeamdynamicsthatarereproducibleinnumericalsimulationsand
experimentsbutdifficultto interpret.Outof thesedegeneratepoints,belowwe consideronly
theMsymmetrypoint.
5.1. X -point
2
InordertomatchtheprofileoftheBlochwavefromtheXsymmetrypointofthesecondband
weuseamodulatedGaussianbeam,oftheform
E(x,y)=Acos[K(x−d/2)]exp(x2/w2+y2/w2),
x y
where the maxima of this modulatedpattern [Fig. 7(a)] are shifted with respect to the lattice
maximabyhalfalatticeperiodalongthexaxis.Thestructureofthedispersionsurfaceofthis
Blochmodeishighlyanisotropic,thereforethebeamdiffractsdifferentlyinxandydirections.
Toaccountforthisanisotropicdiffractionweusedanellipticbeamelongatedalongthexaxis
with w =100µm and w =33µm. At low laser powers(20nW) the beam diffractslinearly,
x y
whilereproducingthestructureoftheBlochwavefromtheXpointofthesecondband(Fig.2).
Withincreaseofthelaserpower[50nW,100nW,and200nWforFig.7(c-e),respectively]the
beamfocusesinbothtransversedirectionsandformsastronglylocalizedstate[Fig.7(e)].Such
state represents the theoretically predicted gap solitons in photoniccrystals [17, 18]. It has a
reducedsymmetrywithrespecttothelatticeanditisformedbythecombinedactionofBragg
reflectioninx-directionandtotalinternalreflectioniny-direction[36].Theexperimentaldata
areinexcellentagreementwiththenumericalsimulations[Fig.7(bottomrow)].
We note that a symmetric superposition of X and Y states gives rise to symmetric gap
2 2
solitons[17,18]orgapvortices[37,38].
5.2. M -point
2
TheMsymmetrypointofthesecondbandisdegenerateasthepropagationconstantcoincides
with that of the third band. Furthermore,two dispersioncurveshave opposite curvaturesand
therefore a complex beam dynamics is expected. We select the Bloch mode [Fig. 2] which