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

-

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

n

2

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.

3

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.

El e c t r oni c addr e s s:yoav.l ahi ni@we i z m ann.ac.i l

[1]P.W.A nde r s on,Phys.R e v.109,1492(1958).

[2]P.A.Le e and T.V.R am akr i s hnan,R e v.M od.Phys.57,

287(1985).

[3]S.John,Phys.R e v.Le t t.53,2169(1984);

[4]P.She ng,I nt r o duc t i on t o W av e Sc at t e r i ng,Lo c al i z at i on,

and M e s os c opi c Phe nom e na,(Spr i nge r,B e r l i n,2006).

[5]J.D.M aynar d,R e v.M od.Phys.73,401(2001).

[6]R.D al i c haouc h e t al.,N at ur e354,53(1991);K.Yu.

B l i okh e tal.,Phys.R e v.Le t t.97,243904(2006).

[7]M.P.VanA l bada and http://www.wendangku.net/doc/f7dbcc6e55270722182ef752.htmlge ndi j k,Phys.R e v.Le t t.55,

2692(1985);P.E.W ol f and G.M ar e t,Phys.R e v.Le t t.

55,2696(1985);E.A kke r m ansand R.M aynar d,J.Phys.

Le t t.,46L1045(1985).

[8]X.Zhang and Z.Q.Zhang,Phys.R e v.B65,155208

(2002);Phys.R e v.B65,245115(2002).

[9]M.V.B e r r y and S.K l e i n She ng,Eur.J.Phys.18,222

(1997);D.S.W i e r s m a e t.al.,N at ur e390,671(1997);

A.A.C habanov e t.al.,N at ur e404,850(2000);M.

St or z e r e t.al.,Phys.R e v.Le t t.96,063904(2006). [10]A.A.C habanov and A.Z.G e nac k,Phys.R e v.E56,

R1338(1997);A.G ar c i a-M ar t i n e t.al.,A pp.Phys.Le t t.

71,1912(1997);Phys.R e v.Le t t.84,3578(2000). [11]A.Tr om be t t oni e t.al.,Phys.R e v.Le t t.88,173902

(2002);K.O.R as m us s e n e t.al.,Eur ophys.Le t t.47,421 (1999);S.A.G r e de s kul and Y.S.K i vs har,Phys.R e p.

216,1(1992);B.D ouc ot and R.R am m al,Eur ophys.

Le t t.3,969(1987);P.D e vi l l ar d and B.Soui l l ar d,J.

St at.Phys.43,423(1986).

[12]G.K opi daki s and S.A ubr y,Phys.R e v.Le t t.84,3236

(2000);Phys i c a D139,247(2000);130,155(1999);C.

A l bane s e and J.Fr ohl i c h,C om m un.M at h.Phys.138,

193(1991)

[13]D.L.She pe l yans ky,Phys.R e v.Le t t,70,1787(1993);

M.I.M ol i na,Phys.R e v.B58,12547(1998);T.K ot-t os and M.W e i s s,Phys.R e v.Le t t.93,190604(2004);

A.P.Pi kovs ky and D.L.She pe l yans ky,ar X i v:0708.3315

(2007);G.K opi daki s e t.al.,ar X i v:0710.2621(2007).

[14]H.S.Ei s e nbe r g,Ph.D.t he s i s,W e i z m ann I ns t i t ut e,2002.

[15]T.Pe r t s c h e t.al.,Phys.R e v.Le t t.93,053901(2004).

[16]T.Sc hwar t z e t.al.,N at ur e446,53(2007).

[17]H.D e R ae dt e t.al.,Phys.R e v.Le t t.62,47(1989);

[18]S.S.A bdul l ae v and F.K h.A bdul l ae v,I z ve s t i ya Vuz.

R adi o z i ka23,766(1980)

[19]D.N.C hr i s t odoul i de s e t.al.,N at ur e424,817(2003).

[20]H.S.Ei s e nbe r g e t.al.,Phys.R e v.Le t t.81,3383(1998).

[21]T he band he r e i si nve r t e d i n r e s pe c tt o t he us ualc onve n-

t i on i n wave gui de l at t i c e s.

[22]A.R.M c G ur n e t.al.,Phys.R e v.B47,13120(1993).

[23]F.M.I z r ai l e v e t.al.,Phys.R e v.E55,4951(1997);E.P.

N ac hm e dov e t.al.,Sov.Phys.JET P65,1202(1987).