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Anderson localization and nonlinearity in one dimensional disordered photonic lattices

a r X i v :0704.3788v 4 [c o n d -m a t .o t h e r ] 7 F e

b 2008

A nder s on l ocal i zat i on and nonl i near i t y i n one di m ens i onaldi s or der ed phot oni c l at t i ces

Yoav Lahi ni 1,A s s af A vi dan 1,Fr anc e s c a Poz z i 2,M ar c Sor e l 2

,R obe r t o

M or andot t i 3,D e m e t r i os N.C hr i s t odoul i de s 4and Yar on Si l be r be r g

1

1

D e p ar t m e nt of Phys i c s of C om pl e x Sys t e m s ,t he W e i z m ann I ns t i t ut e of Sc i e nc e ,Re hov ot ,I s r ae l

2

D e p ar t m e nt of El e c t r i c aland El e c t r oni c Eng i ne e r i ng ,U ni v e r s i t y of G l as g ow,G l as g ow,Sc ot l and 3

I ns t i t ut nat i onalde l a r e c he r c h e s c i e nt i q ue ,U ni v e r s i t e du Q u e b e c ,Var e nne s ,Q u e b e c ,C anada and

4

C R EO L/C ol l e g e of O pt i c s ,U ni v e r s i t y of C e nt r al Fl or i da,O r l ando,Fl or i da,U SA

(D at e d:19A pr i l2007;publ i s he d 10Januar y 2008)W e e xpe r i m e nt al l y i nve s t i gat e t he e vol ut i on of l i ne ar and nonl i ne ar wave s i n a r e al i z at i on of

t he A nde r s on m ode lus i ng di s or de r e d one di m e ns i onalwave gui de l at t i c e s .T wo t ype s of l oc al i z e d e i ge nm ode s , at -phas e d and s t agge r e d,ar e di r e c t l y m e as ur e d.N onl i ne ar pe r t ur bat i ons e nhanc e s l oc al i z at i on i n one t ype ,and i nduc e de l oc al i z at i on i n t he ot he r .I n a c om pl e m e nt ar y appr oac h,we s t udy t hee vol ut i on on s hor tt i m es c al e sof -l i kewave pac ke t si n t hepr e s e nc eofdi s or de r .A t r ans i t i on f r om bal l i s t i c wave pac ke t e xpans i on t o e xpone nt i al(A nde r s on)l oc al i z at i on i s obs e r ve d.W e nd an i nt e r m e di at e r e gi m e i n w hi c h t he bal l i s t i c and l oc al i z e d c om pone nt sc oe xi s tw hi l e di us i ve dynam i c s i s abs e nt .Evi de nc e i s f ound f or a f as t e r t r ans i t i on i nt o l oc al i z at i on unde r nonl i ne ar c ondi t i ons .

T he pr opagat i on ofwave s i n pe r i odi c and di s or de r e d

s t r uc t ur e s ar e at t he f oundat i ons ofm ode r n c onde ns e d-m at t e r phys i c s .A nde r s on l oc al i z at i on i s a ke y c onc e pt ,f or m ul at e d t o e xpl ai n t he s pat i alc on ne m e ntdue t o di s -or de r ofquant um m e c hani c alwave f unc t i ons t hat woul d s pr e ad ove r t he e nt i r e s ys t e m i n an i de al pe r i odi c l at -t i c e [1,2,3,4].A l t hough A nde r s on l oc al i z at i on was s t udi e d e xpe r i m e nt al l y ,t he unde r l yi ng phe nom e na -t he e m e r ge nc e of l oc al i z e d e i ge nm ode s and t he s uppr e s s i on ofwave pac ke t e xpans i on -we r e r ar e l y obs e r ve d di r e c t l y [5,6].I ns t e ad,l oc al i z at i on was us ual l y s t udi e d i ndi -r e c t l y by m e as ur e m e nt s of m ac r os c opi c quant i t i e s s uc h as c onduc t anc e [2],bac ks c at t e r i ng [7,8]and t r ans m i s -s i on [9,10].A n i nt e r e s t i ng i s s ue c onc e r ns t he e e c t of nonl i ne ar -i t y on A nde r s on l oc al i z at i on.N onl i ne ar i nt e r ac t i ons be -t we e n t he pr opagat i ng wave s and nonl i ne ar l y ac c um u-l at e d phas e sc an s i gni c ant l y c hangei nt e r f e r e nc epr ope r -t i e s ,t hus f undam e nt al l y a e c t i ng l oc al i z at i on.T he t he -or e t i c als t udy oft he nonl i ne ar pr obl e m adv anc e d us i ng s e ve r alappr oac he s :t he s t udy oft he t r ans m i s s i on pr ob-l e m [11];t hes t udy oft hee e c tofnonl i ne arpe r t ur bat i ons on l oc al i z e d e i ge nm ode s [12];and t he s t udy oft he e e c t ofnonl i ne ar i t y on wave pac ke te xpans i on i n t he pr e s e nc e ofdi s or de r [13].O nl y a f e w e xpe r i m e nt s we r e r e por t e d [5].R e c e nt l y ,opt i c als t udi e s e nabl e d t he s t udy ofwave e vol ut i on i n nonl i ne ardi s or de r e d l at t i c e s[14,15,16],us -i ng a s c he m edi s c us s e d i n [17,18].I n par t i c ul ar ,Sc hwar t z e t .al .[16]r e por t e d t he obs e r v at i on ofA nde r s on l oc al -i z at i on ofe xpandi ng wave pac ke t si n 2D l at t i c e s .I n t hi s wor k we i nve s t i gat e di r e c t l y l i ne ar and nonl i n-e ar wave e vol ut i on i n one di m e ns i onal(1D )di s or de r e d phot oni c l at t i c e s ,us i ng t wo di e r e nt appr oac he s .I n t he r s t par t oft hi s wor k,al lt he l oc al i z e d e i ge nm ode s ofa we akl y di s or de r e d l at t i c e ar e s e l e c t i ve l y e xc i t e d.N on-l i ne ar i t y i s t he n i nt r oduc e d i n a c ont r ol l e d m anne r ,t o e xam i ne i t s e e c t on l oc al i z e d e i ge nm ode s .T he s e c ond par t of t hi s wor k pr e s e nt s a s t udy of t he e e c t of di s

Anderson localization and nonlinearity in one dimensional disordered photonic lattices

Anderson localization and nonlinearity in one dimensional disordered photonic lattices

Anderson localization and nonlinearity in one dimensional disordered photonic lattices

Anderson localization and nonlinearity in one dimensional disordered photonic lattices

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FI G.1:(c ol oronl i ne ).(a)Sc he m at i c vi e w oft he s am pl e us e d

i n t he e xpe r i m e nt s .T he r e d ar r ow i ndi c at e s t he i nput be am .(b)-(d)I m age s of out put l i ght di s t r i but i on,w he n t he i nput be am c ove r s a f e w l at t i c e s i t e s :(b)i n a pe r i odi c l at t i c e ,(c )i n a di s or de r e d l at t i c e ,w he n t he i nput be am i s c oupl e d t o a l oc at i on w hi c h e xhi bi t s a hi gh de gr e e of e xpans i on,and (d)i n t he s am e di s or de r e d l at t i c e w he n t he be am i s c oupl e d t o a l oc at i on i n w hi c h l oc al i z at i on i s c l e ar l y obs e r ve d.

or de r on t he e vol ut i on of -l i ke wave pac ke t s (s i ngl e s i t e e xc i t at i ons ).A t r ans i t i on f r om f r e e bal l i s t i c wave pac ke t e xpans i on t o e xpone nt i all oc al i z at i on i sobs e r ve d,aswe l l as an i nt e r m e di at e r e gi m e ofc oe xi s t e nc e .W e t he n m e a-s ur e t he e e c t ofnonl i ne ar i t y on t hi s pr oc e s s .O ur e xpe r i m e nt al s e t up i s a one -di m e ns i onal l at t i c e ofc oupl e d opt i c alwave gui de s pat t e r ne d on an A l G aA s s ubs t r at e [19,20],i l l us t r at e d i n Fi g.1a.Li ght i s i n-j e c t e d i nt o one ora f e w wave gui de satt he i nputand c an c ohe r e nt l y t unne lbe t we e n ne i ghbor i ng wave gui de s as i t pr opagat e s al ong t he z axi s .Li ght di s t r i but i on i s t he n m e as ur e d at t he out put (s e e f or e xam pl e Fi g.1(b)-(d)).T hee quat i onsde s c r i bi ng l i ghtdynam i c si n t he s es t r uc -t ur e s ar e i de nt i c al(i n t he l i ne ar l i m i t )t o t he e quat i ons de s c r i bi ng t he t i m e e vol ut i on ofa s i ngl e e l e c t r on i n a l at -t i c e unde rt he t i ghtbi ndi ng appr oxi m at i on [19],i .e .a s e t ofc oupl e d di s c r e t e Sc hr odi nge re quat i ons : i

@U

Anderson localization and nonlinearity in one dimensional disordered photonic lattices

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Anderson localization and nonlinearity in one dimensional disordered photonic lattices

FI G.2:(c ol oronl i ne).M e as ur e m e nt sofl oc al i z e d e i ge nm ode sofa di s or de r e d l at t i c e.(a)C al c ul at e d e i ge nm ode sand e i ge nval ue s oft he we akl y di s or de r e d l at t i c e us e d i n t he e xpe r i m e nt s.T he band ofe i ge nval ue sde vi at e ss l i ght l y f r om t he c os i ne-s hape d band ofa pe r i odi c l at t i c e.Loc al i z e d m ode s ar e f or m e d,as s oc i at e d w i t h e i ge nval ue s ne ar t he e dge s of t he band(i ns e t s1,2),w hi l e m ode s ne ar t he band c e nt e rr e m ai n e xt e nde d(i ns e t3).(b)M e as ur e m e nt s ofpur e at-phas e A nde r s on l oc al i z e d m ode s.Pane l s s how a c om par i s on be t we e n m e as ur e m e nt s(bl ue)and t he c or r e s pondi ng c al c ul at e d e i ge nm ode s oft he l at t i c e(r e d).(c)Sam e f or s t agge r e d l oc al i z e d e i ge nm ode s.I n al lc as e s no t t i ng pr oc e dur e s ar e us e d.

(wave gui de s),U n i s t he wave am pl i t ude at s i t e n, n i s t he e i ge nv al ue(pr opagat i on c ons t ant)as s oc i at e d w i t h t he n’t h s i t e,C n;n 1ar e t he t unne l l i ng r at e s be t we e n t wo adj ac e nt s i t e s,and z i s t he l ongi t udi nals pac e c oor-di nat e.T he l as t t e r m i n Eq.(1)de s c r i be s t he nonl i ne ar de pe nde nc e oft he r e f r ac t i ve i nde x on t he l i ghti nt e ns i t y, w he r e i s t he K e r r nonl i ne ar par am e t e r,w hi c h i s pos i-t i ve f or our s ys t e m( >0):T he nonl i ne ar t e r m c an be di s c ar de d f orl ow l i ghti nt e ns i t i e s.f ort ypi c ale xpe r i m e n-t alpar am e t e r ss e e f or e xam pl e[20].

D i s or de r i s i nt r oduc e d t o t he l at t i c e by r andom l y c hangi ng t he w i dt h ofe ac h wave gui de i n a ni t e r ange W w he r e W i st he m e an v al ue(t ypi c al l y4 m i n our s am pl e s).T he par am e t e r s n and C n;n 1c an be c al c u-l at e d num e r i c al l y f r om t he wave gui de s’w i dt h and f r om t he s e par at i on be t we e n ne i ghbor i ng wave gui de s.A sa r e-s ul tofdi s or de rt he par am e t e r s n be c om e r andom i n t he r ange 0 .By ke e pi ng t he l at t i c e pe r i odi c on ave r age (t he s i t e’s c e nt e r s s t i l lhave t he l at t i c e pe r i odi c i t y),t he par am e t e r s C n;n 1be c om e i nde pe nde nt of n t o a ve r y good appr oxi m at i on,m e e t i ng t he c ondi t i onsas s um e d by A nde r s on i n hi sor i gi nalm ode l(di agonaldi s or de r)[1].A m e as ur e ofdi s or de ri s t he n gi ve n by t he r at i o =C[4]. W e now t ur n t o t he r s t s e t ofe xpe r i m e nt s,de s i gne d t o e xc i t eand m e as ur ei ndi vi dual l y al lt hel oc al i z e d e i ge n-m ode s of a di s or de r e d l at t i c e,and t o s t udy t he e e c t of nonl i ne ar pe r t ur bat i ons on t he m.For t hi s pur pos e we have f abr i c at e d a l at t i c e w i t h N=99s i t e s and di s-or de r l e ve l =C=1.T he di s or de r e d l at t i c e e i ge nm ode s and e i ge nv al ue s c an be c al c ul at e d by di agonal i z i ng a s e t of N e quat i ons(1)i n t he l i ne ar l i m i t,s ubs t i t ut i ng t he c al c ul at e d v al ue s of n and C n;n 1f or t hi s s pe c i c r e-al i z at i on.R e s ul t s ar e s how n i n Fi g2(a).T he r e s ul t i ng band ofe i ge nv al ue de vi at e sonl y s l i ght l y f r om t he c os i ne-s hape d band ofa pe r f e c t l y or de r e d l at t i c e[19].Se ve r al e i ge nm ode s w i t h e i ge nv al ue s ne ar t he e dge s oft he band be c om ee xpone nt i al l y l oc al i z e d i n s pac e,e ve n t hough t he de ns i t y ofs t at e s ne ar t he band e dge s i s not s i gni c ant l y l owe r f r om t he one at t he band c e nt e r[4].Loc al i z e d e i ge nm ode sne art he bot t om oft he band ar e atphas e d, i.e.t he i rwave f unc t i on’sam pl i t ude i si n-phas e atal ls i t e s (s e e i ns e t1)[21],w hi l e t he l oc al i z e d e i ge nm ode s at t he t op oft he band ar e s t agge r e d,i.e.t he i r wave f unc t i on’s am pl i t udehasa phas e i p be t we e n adj ac e nts i t e s(i ns e t 2).N ot abl y,T he s e l oc al i z e d e i ge nm ode s ar e l oc al i z e d at we l l-s e par at e d r e gi onsi n s pac e,and have a s i m i l arw i dt h ofabout10l at t i c e s i t e s.Ei ge nm ode sne art he band c e n-t e r r e m ai n e xt e nde d i n t he ni t e s i z e d s ys t e m(s e e i ns e t 3).T he s e m ode s w i l lal s o be l oc al i z e d i n an i n ni t e s ys-t e m,but on a m uc h l ar ge rl e ngt h s c al e[22].A s di s or de r ( =C)i s i nc r e as e d,a l ar ge r f r ac t i on of t he m ode s be-c om e s l oc al i z e d w i t hi n t he ni t e l at t i c e.

I n or de r t o e xc i t e l oc al i z e d e i ge nm ode s oft hi s l at t i c e we i nj e c t a w i de be am(c ove r i ng s e ve r all at t i c e s i t e s)at di e r e nt l oc at i ons ac r os s t he l at t i c e.A t s om e i nput po-s i t i ons,we obs e r ve s i gni c ant e xpans i on oft he be am at

0.1

0.2

0.3

0.4

0.1

0.3

0.5

0.7

1

(a)(b)

FI G.3:(c ol or onl i ne).T he e e c t of we ak nonl i ne ar i t y on pur e l oc al i z e d e i ge nm ode s:c r os s-s e c t i ons oft he out put l i ght i nt e ns i t i e s(hor i z ont al axi s)at di e r e nt powe r l e ve l s(ve r t i-c alaxi s),s how i ng t hat(a) at phas e d l oc al i z e d m ode s t e nd t o be c om e m or e l oc al i z e d as nonl i ne ar i t y i s i nc r e as e d,w hi l e (b)s t agge r e d l oc al i z e d m ode s t e nd t o de l oc al i z e.A l lc r os s-s e c t i ons ar e nor m al i z e d t o uni t m axi m um.

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Anderson localization and nonlinearity in one dimensional disordered photonic lattices

FI s s how n ar e nor m al i z e d l at t i c e ave r age s oft he out put l i ght di s t r i but i on,w he n i ni t i al l y a s i ngl e s i t e i s e xc i t e d(s e e di s c us s i on i n t he t e xt).A m e as ur e ofl oc al i z at i on i s gi ve n by t he m e an s quar e di s pl ac e m e nt f r om t he i nput s i t e ,and by t he i nt e ns i t y at t he or i gi n s i t e I o.(a-d)M e as ur e m e nt s i n t he l i ne ar c as e s how t he t r ans i t i on f r om bal l i s t i c t r ans por t t o e xpone nt i al(A nde r s on) l oc al i z at i on as a f unc t i on ofdi s or de r i n:(a) =C=0,(b) =C=1,(c) =C=1:5,and(d) =C= 3.N ot e t he t r ans i t i on

f r om bal l i s t i c t r ans por t i n(a)t o l oc al i z at i on i n(d)t hr ough t he bui l dup of a c e nt r alc om pone nt and t he s uppr e s s i on of t he bal l i s t i c s i de l obe s.I ns e t i n(d)s how s t he l oc al i z e d di s t r i but i on i n s e m i l o

g s c al e,de m ons t r at i ng t he e xpone nt i alt ai l s.(e)-(h): M e as ur e m e nt s oft he s am e l at t i c e s i n t he nonl i ne arc as e,s how i ng t haton ave r age nonl i ne ar i t y t e ndst o i nc r e as e l oc al i z at i on f or i nt e r m e di at e di s or de r l e ve l s(e.g.c om par e(b)and(f)).

t he out put,s i m i l ar i n w i dt h t o t he e xpans i on obs e r ve d i n a pe r i odi c l at t i c e w i t h t he s am e ave r age par am e t e r s (s e e g1(b)and1(c)).A t s pe c i c l oc at i ons howe ve r, wave e xpans i on i ss uppr e s s e d and pr om i ne ntl oc al i z at i on i s e vi de nt(Fi g1(d)).I n t he s e c as e s,t he i nput be am ove r l ap s i gni c ant l y w i t h a s i ngl e l oc al i z e d e i ge nm ode of t he l at t i c e,w hi l e t he ove r l ap w i t h t he ot he rm ode soft he l at t i c e i s e l i m i nat e d.O pt i m i z e d l oc al i z e d out put di s t r i-but i on we r e ac hi e ve d us i ng an i nput be am t hat c ove r s about10l at t i c e s i t e s,c or r e s pondi ng t o t he t ypi c alw i dt h oft he l oc al i z e d e i ge nm ode si n t hi s di s or de rl e ve l.

W hi l e s c anni ng t he i nput pos i t i on we obs e r ve s e ve r al l oc al i z e d l i ght di s t r i but i ons at t he out put.W e c om par e t he obt ai ne d l oc al i z e d i nt e ns i t y pr o l e s t o c al c ul at e d l o-c al i z e d e i ge nm ode s of t he l at t i c e i n Fi g2(b).T he r e i s a c l e ar c or r e s ponde nc e be t we e n t he e xpe r i m e nt al l y ob-s e r ve d l oc al i z at i ons and t he l oc at i on and s hape of al l t he c al c ul at e d at-phas e d l oc al i z e d e i ge nm ode s as s oc i-at e d w i t h t he bot t om oft he e i ge nv al ue band.To e xc i t e t he s t agge r e d m ode sas s oc i at e d w i t h t he t op oft he band, t he i nput be am was t i l t e d w i t h r e s pe c t t o t he l at t i c e t o i nduc e a phas e di e r e nc e i n t he e xc i t at i on ofadj ac e nt wave gui de s[19].T her e s ul t soft hi se xc i t at i on s c he m ear e pr e s e nt e d i n Fi g.2(c).A gai n,a c l e ar c or r e s ponde nc e i s f ound t o t he c al c ul at e d l oc al i z e d s t agge r e d m ode s oft he l at t i c e.T he s e r e s ul t s de m ons t r at e t he abi l i t y t o e xc i t e pur e A nde r s on l oc al i z e d e i ge nm ode s.

T he e e c t of nonl i ne ar pe r t ur bat i ons on l oc al i z e d e i ge nm ode s i s s t udi e d by e xc i t i ng a pur e l oc al i z e d m ode and i nc r e as i ng t hei nputbe am powe r.T hei nt e ns i t i e sar e ke ptway be l ow t hos e r e qui r e d t o e xhi bi ts e l f-f oc us i ng i n a pe r i odi c l at t i c e w i t h t he s am e ave r agepar am e t e r s[20], ke e pi ng t he e xpe r i m e nt s i n t he we ak nonl i ne ar r e gi m e. Som e l oc al i z e d m ode sar e f ound t o e xhi bi ts i gni c antr e-s pons e t o nonl i ne ar i t y.T he r e s ul t s oft wo s uc h e xpe r i-m e nt s ar e s how n i n Fi g.3,s how i ng t hat we ak pos i t i ve nonl i ne ar i t y t e ndst o f ur t he rl oc al i z e at-phas e d l oc al i z e d m ode s,but t e nds t o de-l oc al i z e s t agge r e d m ode s.

T he s er e s ul t sc an beunde r s t ood on t hebas i soft het he-or y de ve l ope d i n[12],w hi c h have s how n t hat nonl i ne ar s hi f t s ofa l oc al i z e d e i ge nm ode’s f r e que nc y(r e pr e s e nt e d he r e by t he par am e t e r ),c an l e ad t o de l oc al i z at i on i f t he f r e que nc y c r os s e s a r e s onanc e w i t h ot he r m ode s of t he l at t i c e.T hi s c ondi t i on c an be s at i s e d i n t he c as e of we ak di s or de r f or t he s t agge r e d l oc al i z e d m ode s,as nonl i ne ar i t y s hi f t s t he m f r om t he e dge oft he band i nt o t he l i ne ar s pe c t r um.C onve r s e l y,t he at-phas e d m ode s att he ot he re dge oft he band ar e s hi f t e d by nonl i ne ar i t y out of t he l i ne ar s pe c t r um,t hus t he y r e m ai n l oc al i z e d (s e e al s o di s c us s i on i n[15]).

W e now t ur n t o t he s e c ond s e t ofe xpe r i m e nt s,ai m e d t o s t udy t he e e c tofdi s or de ron wave pac ke te vol ut i on i n t he l i ne ar and nonl i ne ar r e gi m e s.T hi s as pe c t i s s t udi e d by i nj e c t i ng l i ght i nt o a s i ng l e l at t i c e s i t e,t hus e xc i t i ng a t i ght -l i ke wave pac ke t ofal le i ge nm ode s havi ng non-v ani s hi ng ove r l ap w i t h t he e xc i t e d s i t e.T he wave pac ke t t he n e vol ve s i n t he l at t i c e,and t he l i ght di s t r i but i on i s m e as ur e d att he out put.W e ave r aget he out putpat t e r ns obt ai ne d,by s e par at e l y e xc i t i ng e ac h s i t e i n t he l at t i c e w hi l e ke e pi ng t he m e as ur e m e nt-w i ndow c e nt e r e d about t he i nput s i t e pos i t i on.T he r e s ul t s of s uc h m e as ur e-m e nt s i n t he l i ne ar r e gi m e,t ake n i n5m m l ong s am pl e s

4

w i t h di e r e nt di s or de r l e ve l s,ar e s how n i n Fi g.4(a-d). W i t hout di s or de r,s i ngl e s i t e e xc i t at i on r e s ul t s i n bal-l i s t i c pr opagat i on(wave pac ke tw i dt h gr ow sl i ne ar l y w i t h t i m e),r e c ogni z e d by a c har ac t e r i s t i cs i gnat ur eoft wo s e p-ar at e d l obe s[19,20](Fi g.4(a)).A t m ode r at e di s or de r, a s e c ond c om pone nte m e r ge s,l oc al i z e d ar ound t he i nput s i t e pos i t i on(Fi g.4(b,c)).T he l oc al i z e d and t he bal l i s-t i c c om pone nt s c oe xi s t i n t hi s r e gi m e.A t hi gh di s or de r a hi ghl y l oc al i z e d,e xpone nt i al l y de c ayi ng di s t r i but i on i s obs e r ve d(Fi g.4(d)).T hi s e xpone nt i alde c ay oft he e x-pans i on pr o l e i s a hal l m ar k ofA nde r s on l oc al i z at i on. T he s e r e s ul t so e ra r s tdi r e c tl ook att he s hor tt i m e e vol ut i on of wave pac ke t s i n1D di s or de r e d s ys t e m s.I t i s know n t hat f or i n ni t e di s or de r e d1D s ys t e m s and f or l ong t i m e s c al e s,wave pac ke t e xpans i on i s al ways f ul l y s uppr e s s e d.H owe ve r,on s hor t t i m e s c al e s,wave pac ke t s do e vol ve[17,23].T he r e s ul t s i n Fi g.4(a-d)r e ve al how l oc al i z at i on i s r e ac he d i n t he1D c as e;l oc al i z at i on e m e r ge s f r om bal l i s t i c e xpans i on t hr ough t he c ont i nue s bui l dup ofa l oc al i z e d c om pone ntand t he s uppr e s s i on of a bal l i s t i c c om pone nt.T hi s dynam i c s i s f undam e nt al l y di e r e nt f r om t he one obs e r ve d i n2D s ys t e m s[15,16], ort hatwasi ndi r e c t l y m e as ur e d i n quas i-1D e xpe r i m e nt s [10].I n t he s e c as e s t he e xpans i on t ur ns qui c kl y f r om bal l i s t i c t o di us i ve,and be c om e s l oc al i z e d af t e r m uc h l onge r pr opagat i on t i m e s.I n t he1D c as e t he di us i ve dynam i c s i s abs e nt,as di s c us s e d f or e xam pl e i n[23].

To s t udy t he e e c t ofnonl i ne ar i t y on wave pac ke t e x-pans i on we r e pe att he s e m e as ur e m e nt sati nc r e as e d l i ght i nt e ns i t i e s.A gai n,we r e m ai n i n t he we ak nonl i ne ar r e gi m e.R e s ul t s ar e s how n i n Fi g.4(e-h).O n ave r age, t her e s ul t si ndi c at ei nc r e as e d l oc al i z at i on ati nt e r m e di at e di s or de r l e ve l s;t he i nt e ns i t y i n t he bal l i s t i c c om pone nt i s l owe r and t he i nt e ns i t y i n t he l oc al i z e d c om pone nt i s hi ghe runde rnonl i ne arc ondi t i ons.T hi ss ugge s t st hatt he bui l dup oft he l oc al i z e d c om pone nt and t he s uppr e s s i on oft he bal l i s t i c c om pone nthappe n f as t e runde rnonl i ne ar c ondi t i ons.T hi sde s c r i pt i on hol dsf ors hor tt i m es c al e si n w hi c h t he bal l i s t i c c om pone nt i s s t i l lpr e s e nt.O n m uc h l onge rt i m es c al e s,s ubdi us i vede l oc al i z at i on duet o non-l i ne ar i t y waspr e di c t e d t o t ake ove r[13].

I n c onc l us i on,we havedi r e c t l y s t udi e d l oc al i z e d e i ge n-m ode s and wave pac ke t e xpans i on i n di s or de r e d1D l at-t i c e s t hat ar e de s c r i be d by t he nonl i ne ar ve r s i on oft he A nde r s on m ode l.T wo t ype sofl oc al i z e d e i ge nm ode swe r e m e as ur e d,and nonl i ne arpe r t ur bat i onswe r es how n t o e n-hanc e sl oc al i z at i on i n one t ype and i nduc e de l oc al i z at i on i n t he ot he r.T he s t udy oft he e xpans i on ofwave pac k-e t s on s hor t t i m e s c al e s i n t he pr e s e nc e ofdi s or de r has e nabl e d a di r e c tm e as ur e m e ntoft he t r ans i t i on f r om bal-l i s t i c wave pac ke t e xpans i on t o e xpone nt i all oc al i z at i on.

I t was s how n t hat i n1D s ys t e m s a bal l i s t i c and a l o-c al i z e d c om pone nt s c an c o-e xi s t at i nt e r m e di at e t i m e s, w hi l edi us i vewave pac ke te xpans i on,obs e r ve d i n2D and quas i-1D s ys t e m s,i sabs e nti n t he1D c as e.I n t henonl i n-e ar r e gi m e,e vi de nc e i s f ound f or a f as t e r t r ans i t i on i nt o l oc al i z at i on unde r nonl i ne arc ondi t i ons.

W e t hank Y.I m r y,M.A i z e nm an and H.S.Ei s e nbe r g f or us e f uldi s c us s i ons.T hi s wor k was s uppor t e d by t he G e r m an-I s r ae l iPr oj e c tC oope r at i on(D I P),N SERC and C I PI(C anada),and EPR SC(U K).Y.L.i ss uppor t e d by t he A dam sFe l l ow s hi p Pr ogr am oft he I s r ae lA c ade m y of Sc i e nc e s and H um ani t i e s.

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