Nov 5, 2003 - Page 1. Vol. 105 (2004). ACTA PHYSICA POLONICA A. No. 4. Light Propagatio n in a Magn eti c Fie ld: Random Green Matri x Ap proach.
V ol . 1 0 5 ( 2004)
A CT A P HY SIC A P O LO N IC A A
No . 4
L igh t P r op agatio n in a Magn eti c Fie ld : R an d om G re en Matri x Ap p roach F .A . P inhei r o a , M. R usek b , A . Or ¤o wsk i b and B.A . van Ti gge l en a a
CN R S/ L ab orato i r e de P hysique et Mo d Ç eli sati on des Mi l i eux Co ndensÇ es Uni versi tÇ e Joseph Fouri er, Ma i son des Ma gi st ˚eres B. P. 166 38042 G reno bl e Cedex 9, Fra n ce b
Instytut
Fi zy ki , Po l ska Ak ademi a Na uk
al . Lo tni k§w 32/ 46, 02- 668 Warsza wa , Po l and
( Recei ved N ovember 5, 2003; revi sed versi on Januar y 20, 200 4) W e studied the spect ral pr op erties of the matrices descri bing multiple scattering of electromagneti c w aves from randoml y distribu ted point -li ke magneto- optical l y active scatterers under an external magnetic Ùeld B . W e show ed that the complex eigenv alues of these matrices exhibit some universal prop erties such as the self-a veraging behavior of their real parts , as in the case of scatterers without magneto- optical activity . H owever, the presence of magneto- optical l y active scatterers is resp onsib le for a striking particulari ty in the spectra of these matrices : the splitti ng of the v alues of the imagina ry part of their eigenvalues. T his splitting is prop ortion al to the strength of the magnetic Ùeld and can be interpreted as a consequence of the Zeeman splittin g of the energy levels of a single scatterer. PAC S numb ers: 42.25.Dd, 78. 20.Ls
1. I n t r o d u ct io n W a ve pro pa gati on i n compl ex m edia i s a bro ad and i nterdi scipl i na ry research to pi c, wi th many appl i cati ons [1{ 4]. Medi a exhi bi ti ng bro ken sym m etry such as ma gneto- opti cal and/ or chi ra l m edia, no vel di sordered m ateri al s such as stro ngl y scatteri ng semi conducto r p owders, and na tura l m edia such as the subsurface of the Ea rth are j ust a few exampl es of com pl ex m edia in whi ch wa ve pro pagati on ha s b een recentl y studi ed. In parti cula r, the presence of m agneto- opti cal (3 39)
340
F.A . Pi n heiro et al .
acti vi ty , associated wi th the brea ki ng of ti m e-reversal sym metry and very well studi ed i n ho mogeneous medi a [5], was shown to be at the ori gi n of a new class of opti cal pheno mena i n i nho m ogeneous m edi a. For i nsta nce, the suppression of coherent backscatteri ng e˜ect by the Fara day ro tati on was extensi vel y studi ed, b oth experi m enta l l y [6{ 8] and num eri cal ly [7{ 9]. Theo reti cal l y, the i nÛuence of m agneto -opti cal acti vi ty on the coherent ba ckscatteri ng cone was i nvesti gated by Ma cKi nto sh and John [10], van Ti ggelen et al . [11] and La coste and van Ti ggel en [1 2]. It was theoreti call y suggested [13] tha t the Fara day e˜ect generates a m agneto -tra nsverse l i ght di ˜usi on in ra ndo m m edia sim i lar to the electro ni c Hal l e˜ect. Thi s so-call ed pho toni c Ha l l e˜ect, whi ch Ùnds its ori gi n in l i ght scatteri ng by one sing l e Fara day-a cti ve scatterer , was recentl y observed [14, 15]. In spi te of these extensi ve studi es o ver the l ast years, the i m pact of m agneto -opti cal e˜ects o n some op en and funda menta l pheno m ena i s sti l l unkno wn. The study of the onset of Anderso n or stro ng l o cal i zati on of l ig ht i s one of them . Anderso n l ocal izati on ref ers to an i nhi bi ti on of wa ve tra nsport i n di sordered medi a due to the i nterf erence of m ul tipl e scattered wa ves [16]. Anderso n l ocal izati o n of l i ght was recentl y rep orted in semi conducto r powders [17] and di sagreement even exi sts wi tho ut mag neti c Ùelds [18]. As an al terna ti ve, i t was real i zed tha t cl ouds o f l aser-cool ed ato m s consti tute a pro mi sing scatteri ng m edi a i n whi ch Anderso n l ocal i zati on of l i ght coul d b e achi eved [19]. In f act, they consti tute a p erfectl y m ono di sperse and non- absorbi ng system of resonant p oi nt- li ke scatterers wi th l arg e cross-sectio ns and hug e magneto -optica l e˜ects [20]. A deeper understa ndi ng of the onset of the lo cal i zed regi me i n such medi a, i ncl udi ng the ro l e of m agneto -opti cal acti vi ty , i s thus needed and thi s consti tutes the m ai n m oti vati on for the present work. Thi s m oti vati on was sti mula ted by the Ùrst exp eri ments on m ulti pl e l i ght scatteri ng i n col d atom i c clouds [21], i ncl udi ng the observ ati o n of the coherent ba ckscatteri ng e˜ect i n col d rubi di um [21] and stro nti um [22]. Esp ecial ly for stro nti um , mi m icking a two -l evel a tom near the frequency where the exp eri ments ha ve b een conducted, shoul d be very wel l descri b ed by the present study . W e wi l l i nvesti gate electro m agneti c wave pro pagati on i n m agneto- optica l l y acti ve m edia wi thi n the fra mework of the ab i ni ti o appro ach i ntro duced earl ier by R usek and Or ¤owski [23]. Thi s appro ach i s ba sed on a study of the spectra of the G reen ma tri ces descri bi ng wa ve scatteri ng f rom an assembl y of rando m l y di stri buted p oi nt- l i ke di p ol es (i .e., pa rti cles m uch sm al ler tha n the wa velength of l i ght), and has b een empl oyed to i nv esti gate Anderso n l ocal i zati on of scal ar wa ves i n the case of a 2D system [24] wi th nontri vi a l b ounda ry condi ti ons [25]. It was shown tha t the spectra of these ra ndo m G reen matri ces exhi bi t som e uni versal pro perti es, such as the \ clusteri ng " of the real pa rt of thei r eigenvalues emerg ing i n the l i mi t of a n i nÙnite m edium , whi ch wa s interpreted as the app eara nce of l ocal i zed sta tes [23, 26]. In the present pap er we m odi fy thi s metho d in order to i ncl ude the m agneto- opti cal acti vi ty i nside the di p ol es.
Li ght Pro pagati on i n a Magnet i c Fi el d
341
. ..
2 . T - m at r i x i n t h e p r esen ce o f a m a gn et i c Ùel d In the fol l owing , we deal wi th classical magneto -opti cal di p ol es l ocated i n vacuum . The 3 È 3 T -m atri x t ( k ) descri b es the l i ght scatteri ng by one pa rti cl e and exhi bi ts a scatteri ng resona nce at frequency ! 0 wi th li ne wi dth À ( k = ! = c 0 i s the wa ve num b er) [11]. Al l multi ple scatteri ng pro cesses i n an assembl y of N m agneto -opti cal scatterers situa ted at the p ositi ons r 1 ; r 2 ; . . . ; r N are ful l y descri bed by the 3 N È 3 N M m atri x [23{ 27]: M
(k)
≤
T (k )
Â
[U
G( k ) Â T ( k ) ]
À
À
1
(1 )
;
where U i s the 3 N È 3 N uni t m atri x, T (k ) i s a 3 N È 3 N m atri x whi ch ha s 3 È 3 di agonal bl ocks equal to t ( k ) and zeroes otherwi se, and the 3 È 3 bl ocks of the 3 N È 3 N G -m atri x are gi ven by n 8 ex p ( ik r ) > À (U À b rnm b rnm ) > i 4 ¤ r < ± ≤ o 1 1 G n m (! ) = (2 ) À + [U À 3 b r r for n 6= m; n m b n m ] ik r (kr ) > > : 0 for n = m n m
nm
nm
nm
(m; n = 1 ; 2 ; . . . N ). G n m may b e understo o d as an el ectri c G reen functi on of the Ma xwel l equati ons cal cul ated fro m the rel ati ve positi ons of the N di p ol es [28]. The 3 È 3 to ta l T -m atri x i s gi ven by XN T
XN e
À
=
;
m =1
i
Â
M mn ei
Â
(3 )
;
n =1 0
where M mn i s the mn - th 3 È 3 bl ock of the 3 N È 3 N matri x M , and k a nd k are, respecti vel y, the i nci dent and the scattered wa ve vecto rs, and i n the far Ùeld j k j = j kj = k = ! = c0 . Equa ti on (3 ) determ ines the el ectri c Ùeld scattered by the m agneto- opti cal di po les. It requi res the di a gonal i zati on of the M -m atri x (1 ). The t -m atri x descri bi ng one singl e scatterer t ( ! ) can b e expressed as a Bo rn series [28]: 0
ç t(!
) =
V(!
V(!
) +
)
Â
G 0( !
)
Â
V(!
Ñ
À
1 ) + .. . =
V
À
(! )
G 0( !
)
1
;
(4 )
where V (! ) i s the opti cal p otenti al for a p oint scatterer and G 0 ( ! ) i s the return G reen functi on, i .e., the dya di c G reen functi on G 0 ( ! ; r ) eval uated at r = 0. The opti cal p otenti al i s, i n contra st to the case of qua ntum m echani cal scatteri ng theo ri es, frequency- depend ent and for a di pol e at the po siti on r i subj ect to an externa l m agneti c Ùeld B i t ta kes the form ˚ Ç2 V
(! ;
r)
=
!
À
c0
˜ ( B )£ ( r
À
ri
);
(5 )
342
F. A . Pi nhei ro et al .
where ˜ ( B ) i s the pol ari zabi l i ty tenso r of the scatterer. It has the di mension of a vo l ume whi ch represents the mi croscopi c size of the di p ol es and i s typi cal l y of the order of a 30 ( a 0 | the Bo hr ra di us), but can b e much bi gger when the R ydb erg sta tes are inv ol ved i n the tra nsi ti on. For a m agneto -opti cal di p ol e the p ol ari zabi l it y ˜ ( B ) i s gi ven by U
1 =
˜ (B)
+ ˜
( B );
L
(6 )
0
where ˜ 0 i s the ordi na ry pol ari zabi l i ty of the scatterer [29 ] and L ( B ) i s a m agneto - opti cal correcti on. We shal l restri ct ourselves to l i near orders in B , i n whi ch case thi s correcti on i s gi ven by [11]: L
ij
( B ) = Ù( B ) i¯ i j
k
Bbk ;
(7 )
where i ¯ i j k Bb k i s an anti sym metri c Herm iti an tenso r i n term s of the Levi -Ci vi ta tenso r ¯ i j k ; B = j B j , and Bb = B = B . The di m ensionl ess qua nti t y Ù ( B ) is deÙned ± ≤2 by Ù ≤ ˜4 ¤ !c VÀB ! c , wi th V the V erdet consta nt of the scatterer [11]. 3
0
3 0
0
Equa ti on (2 ) shows tha t the m atri x elements of the return G reen functi on G 0 ( ! ) exhi bi t singul ari ti es at r = 0 . Thi s feature can b e ha ndl ed by regul ari zati on [29]: ˚ Ç ~ 0 (k; r G
Ê = 0) =
+
k
i
6¤
U:
(8 )
6¤
The i nverse l ength scal e Ê deÙnes ! 0 and À by the rel ati ons 1 = Ê = ( ! 0 = c) 2 ( ˜ and 1 = Ê = ( À c= ! 20 ) , respecti vely. Usi ng Eqs. (8 ) and (4 ), we can wri te ˚ Ç 2 2 t
À
1
(! ) =
V
À
1
(! )
À
G 0 (!
) =
Ê
1
!
À
6¤
!
0
U
2
+
i
!
c
À
6¤ c
!
2 0
L
(B);
0 =6¤
)
(9 )
where Eqs. (5 ), (6 ) ha ve b een used. Assum i ng tha t ! 0 À § 1 (f or an ato m , typi cal l y ! 0À ¤ 1 0 6 ) we can put, close to the resonance, ! ¤ ! 0 , and i ntro duce the detuni ng  ≤ (! À ! 0 )= À . W e can rewri te the to ta l scatteri ng m atri x M ( ! ) i n Eq. (1) as ç Ñ 1 À
À
M
4 ¤ c0
(! ) =
!
UÂ
+
2
Ui
3
0
4¤ c 0
À
!
G(!
0
)
À
L
(B )
:
(1 0)
0
Let us no ti ce tha t the onl y frequency dependence of the to ta l scatteri ng matri x (1 0) near the resona nce i s conta i ned i n  . Equa ti on (1 0) shows tha t to Ùnd the T - matri x i n Eq. (3 ), we need to di agona l ize the m atri x GMO (!
0)
=
4¤ c !
G( !
0
) +
L
(B):
(1 1)
0
Thi s m atri x i s in d e pend ent of the frequency ! and the presence of the second term i n (1 1) m akes the di ˜erence wi th previ ous works [23{ 27]. In the fol l owi ng section, we wi l l num eri cal ly di agona l ize the matri x G M O for an arbi tra ry numb er of scatterers and i nvesti gate the consequences of the m agneto -opti cal acti vi ty for the di stri buti on of i ts eigenvalues.
Li ght Prop agati on i n a Magnet i c Fi el d
343
. ..
3 . R esu l t s an d d i scu ssio n As an exam pl e of a typi cal di stri buti on of the ei genvalues Ñ of the rando m l y chosen G M O m atri x, in Fi g. 1 we pl ot the real and the i m agina ry pa rts of the spectrum o f G M O for a conÙgura ti on of N = 1 0 0 0 resona nt di pol es i n a sphere, wi th the uni form density £ = 1 scatterer p er wa vel ength cubed Ñ30 . Thi s situa ti on resembl es the case of scal ar scatterers [23{ 27]. W e observe tha t eig envalues tend to \ cluster" them selves aro und the li ne ReÑ = À 1 . Previ ous num eri cal wo rk ha s shown tha t thi s ki nd of beha vi or app ears by i ncreasing the size of the system [23{ 27]. Thi s f act suggests tha t i n the l i mi t of an i nÙni te system al l the eigenval ues wi l l m ove to ward the value R eÑ = À 1 , a tra nsiti on tha t m ay b e i nterpreted wi th the form atio n of a band of l ocal ized sta tes [23{ 27].
Fig. 1. N
=
Sp ectrum of the G M
1 000
O
matrix
corresp ondi ng to an arbitrary
conÙguration
of
point- like scatterers calcula ted w ith a v anishing v alue of the external mag-
netic Ùeld. T his situation
corresp onds to the case of scatterers without
magneto- optical
activity .
The i m pact of the m agneto- opti cal acti vi ty on the spectra of the matri x G M O can b e seen i n Fi g . 2 where we pl ot the real and the i magina ry pa rts of Ñ ( B ) cal culated wi th the same para m eters as i n Fi g. 1 but now wi th a no nzero val ue of the externa l m agneti c Ùeld ( Ù (B ) = 5 ). We can observe no t onl y the tendency of the eigenva lues to \ cluster" them selves a round the li ne R eÑ = À 1 , as i n the case of scatterers wi tho ut magneto -opti cal acti vi ty , but al so a stri ki ng consequence of the m agneto -opti cal acti vi ty: the spl itti ng of the val ues of the i m agi nary part Im Ñ of the eigenval ues. In order to expl ai n thi s result, one shoul d recal l the physi cal m eani ng of the real and i magina ry pa rts of the eigenvalues o f the G M O m atri x. W i thi n our m odel for each sing l e scatterer, whi ch exhi bi ts a n i nterna l scatteri ng resona nce (a l so cal l ed the Brei t{ W i gner typ e resonance), the real and i m agi nary pa rts of the eigenvalues corresp ond appro xi m atel y to the rel ati ve wi dth and to the
344
Fig. 2.
F. A . Pi nhei ro et al .
A s in Fig. 1, but now w ith a non- v anishing v alue of the external magnetic Ùeld.
T he v alue of the dimension less magnetic Ùeld strength us notice the splitting
of the v alues of the imaginary
absent in the case of scatterers without
Fig. 3.
parameter
is
Ù(B ) =
magneto- optical activity
Ù(B ) =
10 :
Let
exhibited in Fig. 1.
A s in Fig. 2, but now w ith a value of the magnetic Ùeld strength
tw ice larger, i. e.,
5: 0.
parts of the eigenv alues, w hich is
parameter
.
p ositi ons of the resona nces, respecti vel y [27]. As a result, the observed spli tti ng of the val ues of Im can b e i nterpreted as a consequence of the Zeem an spli tti ng of the energy l evels of each scatterer i nduced by the appl i cati on of an externa l m agneti c Ùeld. Thi s i nterpreta ti on i s conÙrm ed by the results shown i n Fi g. 3, where we ha ve cal cul ated the spectra o f the m atri x for the same numb er of scatterers and pa ram eters used i n the previ ous Ùgures but no w wi th a m agneti c Ùeld streng th pa ra meter twi ce la rger tha n i n Fi g . 2 , i .e., . By compa ri ng the
Li ght Prop agati on i n a Magnet i c Fi el d
. ..
345
p ositi ons where the val ues of Im Ñ tend to di stri bute them selves wi th the ones i n Fi g. 2, we noti ce tha t they are e˜ecti vely l i near ly prop ort i onal to the m agneti c Ùeld streng th para m eter Ù ( B ) . Thi s fact conÙrm s tha t the observed spli tti ng of the val ues of Im Ñ actua l l y reÛects a genui ne Zeema n spli tti ng of the energy levels of a singl e scatterer. In addi ti on, we observe tha t for a l arg er val ue of the m agneti c Ùeld, the di stri buti on of the val ues of Im Ñ aro und thei r spl i t p ositi ons tends to b e l ess pro no unced.
Fig. 4. Surf ace plot of the density of eigenv alues P ( Ñ ) calculated for 100 distinct conÙgurations of N = 300 p oint- like scatterers. T he value of the magnetic Ùeld strength parameter is
Ù (B ) = 5
.
In Fi g. 4 we pl ot the pro ba bi l i ty di stri buti on P ( Ñ ) cal cul ated fro m 100 di sti nct real izati ons of the di sorder of 300 scatterer s di stri buted i n a sphere wi th uni form density £ = 1 and subj ect to an externa l m agneti c Ùeld. The pro ba bi l i ty di stri buti on P ( Ñ) was no rm al i zed by d ÑP ( Ñ ) . The i nspecti on of thi s pl ot reveal s tha t val ues of Im Ñ are appro xi matel y eq u ipar titi oned b etween the three spli t positi ons i n the Im Ñ a xi s i nduced by the externa l m agneti c Ùeld. Furtherm o re, P ( Ñ ) tends to m o ve towa rds the R eÑ = 1 l ine, as observed i n the case of scatterers wi tho ut m agneto -opti cal acti vi ty [27]. Thi s \ clusteri ng " b ehavi or of the spectra of the rando m Green m atri ces, whi ch we no w also veri fy when the form of these m atri ces i s mo di Ùed by the presence of an externa l magneti c Ùeld, seems to be, as p ointed out by Rusek et al . [27], a trul y uni versa l beha vi or for ra ndo m G reen m atri ces f or whi ch som e simpl e underl yi ng expl anati on arg um ent shoul d exist.
In conclusi on, we ha ve i nvesti gated the pro perti es of the m atri ces tha t descri b e m ul tipl e scatteri ng of electro m agneti c wa ves by the Fara day acti ve di p ol es
346
F. A . Pi nhei ro et al .
subj ect to an externa l m agneti c Ùeld. These m atri ces can b e expressed as a sum of the com pl ex-sym metri c ra ndo m G reen matri x, whose elements are equal to the free-space G reen functi on cal cul ated fro m the positi ons of each pa i r of scatterer s, and a bl ock- di agona l matri x, l i nearl y pro p orti ona l to the m agneti c Ùeld, whi ch i s associated wi th the magneto - opti cal correcti on. We ha ve shown tha t the spectra of these m atri ces exhi bi t the sam e uni versal pro p erty as i n the absence of m agneto -opti cal acti vi ty , whi ch i s the \ clusteri ng" of the real pa rt of thei r ei genval ues aro und the l i ne R eÑ = À 1 . Howev er, the presence of an externa l m agneti c Ùeld m odi Ùes the spectra i n a stri ki ng way: i t i nduces a spli tti ng of the val ues of the i magi na ry pa rt of the eigenvalues. Thi s spl itti ng i s pro po rti ona l to the streng th of the m agneti c Ùeld. Physi cal l y, since the i magi na ry part of the spectra appro xi m atel y corresp onds to the l ocati ons of the resona nces of a sing l e scatterer , thi s spl itti ng of the spectra can b e interpreted as a genui ne Zeeman spli tti ng of the energy l evels of a singl e scatterer. W e thi nk tha t results presented here consti tute an i m p orta nt step for the study of the pheno menon of the Anderso n l ocal izati o n of li gh t using the fra m ework of the ra ndo m G reen m atri x metho d. In o rder to unam bi guo usly study the onset of the l ocal i zed regi m e i n medi a subj ect to m agneti c Ùelds, one shoul d i nvesti gate further aspects of l i ght pro pa gati on i n such m edia, such as, for i nsta nce, the pattern of the coherent ba ckscatteri ng cone. The study of the ba ckscatteri ng cone i n such a m edi um can pro vi de som e predi cti ons about the mag ni tude of the tra nsport m ean free path i n the l ocal i zed regi m e. These predi cti ons coul d b e experi m enta lly veri Ùed, for i nsta nce, i n clouds of col d ato ms exhi bi ti ng i mp orta nt m agneto -opti cal e˜ects. These aspects are under inv esti gati on and wi l l b e repo rted soon.
Ac kn owl ed gm en t s Thi s wo rk was supp orted by the Sta te Comm ittee for Scienti Ùc Research (Po l and) under gra nt no . 2 P0 3B 044 19.
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Li ght Prop agati on i n a Magnet i c Fi el d
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