On the problem of optical bistability of nonlinear composites with coated inclusions
The electrodynamical properties of the nonlinear metal composites are intensively studied in many papers [1-7]. One of the most important property of such systems is the abnormal enhancement of the nonlinear optical respond in the composites containing small inclusions (compared to the wavelength of...
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| Datum: | 2001 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine
2001
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|---|---|
| author | Grechko, L. G. Davidova, O. A. Mal'nev, V. N. Whites, K. W. |
| author_facet | Grechko, L. G. Davidova, O. A. Mal'nev, V. N. Whites, K. W. |
| author_institution_txt_mv | [
{
"author": "L. G. Grechko",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "O. A. Davidova",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "V. N. Mal'nev",
"institution": "University of Kentucky"
},
{
"author": "K. W. Whites",
"institution": "University of Kentucky"
}
] |
| author_sort | Grechko, L. G. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-11-27T09:42:39Z |
| description | The electrodynamical properties of the nonlinear metal composites are intensively studied in many papers [1-7]. One of the most important property of such systems is the abnormal enhancement of the nonlinear optical respond in the composites containing small inclusions (compared to the wavelength of radiation) of a nonlinear dielectric covered by the metal shell embedded in a dielectric host matrix [2-6]. The surface plasmons in the metal shell may be tuned in resonance with the external electromagnetic field and produce a considerable increase in the local field in the core of inclusion to make the nonlinear part of its dielectric permittivity to be important. As a result, the connection between the applied and the local field in the core becomes nonlinear and in some diapason of applied electric fields even ambiguous. It happens to be that one value of the applied field corresponds to a few values of the local fields and polarization of the inclusions that in its turns leads to instability in the composite optical properties. This phenomenon is called the intrinsic optical bistability (IOB).
In this paper we calculate the dielectric permittivity and polarizability of a separate inclusion_and analyze the parameters of the IOB. Further, we consider the dielectric function of |
| first_indexed | 2025-09-24T17:44:39Z |
| format | Article |
| fulltext |
oN{TIIЕPROBLЕMoFOPTICALBISTABILITYoF
]\OI\LINЕAR CoМPoSITЕs WITII сOATЕD II\сLUSIONS
L.G. Greсhkol, O.A. Davidovаl, V.N. Мal'nеv2, and K-w. Whitеs2
lInstitutе of Suфсе Сhеmistrу, Nationаl Асаdеmу of Sсiепсеs,
Gеtt. Nаumoу str, ]7, oiвво"куi",-l61, UKRАINЕ: е.mail: usеrфuфhеm,frеenet,kiеv.uа
, Depаrtmепt o.f iIесrriсаl Епgitlееriпg' (lnivеrsitу of Kеntuсkу,
453 Аttdеrson Наlt, iехiпgtoп, KY 10506, (]SА; е-mаil: whitеs@еngr,ulсу,еdu
Abstraсt
Thе еlесtrodуnamiсal рropеrtiеs оf thе nonlinеar mеtal сompositеs aге intеnsively
studiеd in manу рapеrs tl.7]. onе of thе most impoгtant propeгtу of suсh sуstеms is thе
abnormal еnhanсеment oЪ thЬ nonlinеar optiсal геspond in thе сompositеs сontaining small
inсlusions (сomparеd to1hе *uu.lеngth of гadiation) of a nonlinеar-diеlесtгiс сoveгed bу thе
mеtal shеll еmbеddеd in a diеlесtгiс hЪst matгiх [2-6] Thе
'uф9:
plasmons in thе mеtal shеll
maу bе tuned in геsonanсе with thе ехtеrnal еlесtromagnеtiс fiеld and produсе a considеrablе
inсгеasе in thе loсal fiеld in thе сoге of inсlusion to makе thе nonlinеar paп of its diеlесtгiс
pегmittivity to bе impoгtant. As a rеsult, thе сonnесtion bеtwееn thе appliеd and the loсal Гrеld
in thе сorе bесomеs nonlinеaг and in sоmе diapason of applied еlесtriс fiеlds еvеn ambiguous.
It happеns to bе that й uu1u" of thе appliеj fi.1d .o'."'ponds to a fеw valuеs of the loсal
fiеlds and polaгizаtion of thе irrоlusions that in its tuгns leads to instabilitу in the сomposite
"pii..l
p.op"пi"s TЫs phеnomеnоn.is сallеd thе intгinsiс optiсal bistabilitу GoB)
In this pup.. *Ь сalсulatе thе diеleсtгiс peгmittivity and. polaгizability of a sеpaгate
inсlusion.and ana|уzetь. p*u'*.rs of the IoB' Fuпhег, we сonsider the diеlесtriс funсtion of
1. Loсаl fiеld in сoаtеd sрhеre рartiсlе
Lеtusсonsiае,a-щhеriсalinсlusionсovегеdйthamеtаlshеllofanoutеггadiusrz'
Thе сorе of thе inclusion is thе Kеrr tуpе nonlineaг diеlесtгiс of a radius r'' Wе сhoose thе
diеlесtriс funсtiоn оf thе core in thе fогm
€l = €to * xlE,| (1 l)
whеrе sto is a linеar paгt of thе diеlесtгiс funсtion, 1is thе nonlinеar^Kегг сoеffiсiеnt, and.Еr is
an amplitudе of thе localfiеld in thе сorе, Thе diеiЬсtriс fi.rnсtion of thе mеtal shеll lеt bе thе
Drudе tуpе
- ',n (1 2)
a1 -
oс,
a(a+tv)
. :;. с
-. ,: !
-.aru
-. .l>lr
-:i :itl
'. ';. tL]с
} lnсе
)tе e^ (r
,.4 A
. : buik
.;Cuгs
.oг еt
nkoKуivNationalUniveгsitу,PhysiсsDерaгtmеnt'
ProsDесt Glushkova 6' oз 1 42Kyir,, Ukraine; еmail : vmal@phys.univ.kiеv.ua
r68
r-oF
_ sroNs
-hitеs2
:,:еt 'kieу.uа
г: :ntеnsivеly
:.;:зms is thе
г:.ring small
с ' =:еd by thе
:: :lеtal shеll
r :. rsidегablе
г :s ciеlесtriс
.-:. loсal fiеld
r ::biguous.
х : thе loсal
:::cmpositе
П
:, . sеparatе
]т': .-псtiоn of
п:.. :adius 12'
J* ...ооsе thе
)
зnd Еr is
зt bе thе
-r
whеrе s- doеs nоt dеpеnd on frеquеnсу o i щ , aпd v aге the plasma and сollisioп frеquеnсу
оf сonduсting еlесtrons геspeсtivеlу. Thе diеlесtгiс pегmittiйtу of thе host matгix matеrial wе
dеnotе as а.,
Lеt thе eхtегnal varying with timе еlесtrоmagnеtiс fiеld E = Ёoе-'- aсts on thе inсlusion. In
the long rvavеiеngth limit r" ((i Ji'' ,,((9,F) thе distгibution of the еlеоtгiс fiеld within
thе inсlusion maу bе found by solving thе Laplaсе equation foг thе two-layеr sphеre in thе
homogеnеоus сonstant еleсtriс fiеld Еo Bу using thе сontinuity сonditions fог thе еlесtгiс
рotеntial and thе noгmal сomponеnt of thе elесtгiс induсtion on thе inсlusion intегГaсеs onе
сan gеt the lосal fiеld as a funсtion of thе аpplied fiеld
A, = Йo, (l 3)
whеrе 7 is сallеd thе enhanсеmеnt faсtoг and givеn by the following eхpression
9еIу=-.:' 2p A'
ь = (сi). +|(з /2p _\сi +З l p -I]вi + еi
Hеге we usе the геlative diеlесtгiо funсtions with геspесt to s-?, in paгtiсulaг с'':с/с^(i:|'2),
p:1- r iз ,, r2.. is a fraсtion of the mеtal in thе inсlusion, Bеlow, an indеx r will be omittеd if it
dоеs not lеad to сonfusion.
it is сlеaг from (1 3) and (l 4) that onе may obtain a сonsidеrablе incrеase in thе loсal еlесtriс
fiеld pгovidеd that A_+0. It сan bе donе by tuning thе paгameters еnteгing in (1.4). Wе
сonsidеrthе simplеst сasе in the limit, v/al<<|, whеn an imaginary paп of (l 2) is nеgligible.
In this сasе thе сonditiоn оf thе ..resonanсе'' сoггеsponds to A:0 that rеduсеs to a quadгatiс
equation in в2 ( 1 4) Roоts of this еяuation mау bе wгitten in thе form
s,'=(0t.lp, _aс{)l2, B=Ql2p_l)ei+3lp_1 (1 5)
Wеnotеthatinouгсasеalwaуs B<Оandthегoots €,r1o, Thatiswhy с, <я,* andaswell'
In thе limiting сasе of small mеtal fraоtion , p<<1, ехpгеssions ( l '5) maу bе simplified:
(1 4)
(l 6)
Foг
a-
Sinсе e.- aге nеgative the геalization of thе еnhanсement of thе local fiеld rеquiгes that
Rе;'(a-l) will bе nеgativе as wеll, onе сan sее from (1.2) that it takеs plaсe at frequеnсiеs
Ф 1ac = Ф P l Б (wе assumе that 4}>> и) Thе frеquеnсy atl сorгеsponds to the frеquеnсy
of bulk plasmons. it atso follows from (1 2) that a maхimum еnhanсemеnt of the local fiеld
oссurs whеn thе ftеquеnсу of еlесtгomagnеtiс wavе aррroaсhеs to
4,, (l 7)
"2t
еxamplе' at p <<l,
-+ 0, сi:l.. ) aa.
whеn thе mеtal fraсtion of inсlusion is сompaгativеly small,
169
Thе еnhanсеmеnt of loсal fiеld in the inсlusion сorе mеans that the nonlinеаr teгm in
(l.1) must bе takеn into aссount' Thеrеfoге, thе еnhanсеmеnt faсtoг r 0 4) itsеlfdepends on
thе loсal fiеld tr and, in a genегal сasе, rеlation (l.3) tuгns out to bе the equation for thе loсаl
fiеld as a funсtion of appliеd fiеld Еo Insеrting (1.1) into (1.4) and multiplying a new
ехprеssion (1 3) by its сomplеx сonjugatеd, wе obtain thе folloйng сubiс equation foг
; ;. l;12Lt.t't =lrtll
,, * zк.(9l*, *l*.]' х = тlхo''5' l6l
- t2 r- rl
х = I, Е,l- ,0. ," = r,|Еo| ,0, Ao =
^ki
+ а,Ъ)' (l 8)
5 =1+ е\('3l2p-|),тl =+l+l'Z, _ x l €,.4p.1ё
|
This сubiс еquation has геа-l сoеffiсiеnts and may havе one геaI positivе root oг thrее rеal
positivе гoots dеpеnding оn its paramеtеrs It will bе analуzed in dеtail in thе nехt sесtion.
ilo*\I/. would likе to pay moге attеntion to the phуsiсаl mеaning of this rеsult. Appеaranсе of
thrеe diffегеnt valuеs ofthе loсal fiеld that сorrеspond to onе valuе ofthе appliеd еlесtriс fiеld
mеans that thе systеm bесomеs unstablе. Тhis phenomеnon is сalled thе intгinsiс optical
bistability (loB) t6] аnd assoсiatеd with a suddеn оhangе in the optical pгopеrtiеs of thе
inсlusion and thе dispегsе systеm as a wholе dеpеnding on the amplitudе of the inсidеnt
гadiation'
Hеre wе would likе to notе that thе polaгizabilitу of thе two-laуеr spheгiсal inсlusion
maуbeprеsеntеd'"'':"j';l:Цl',
(1 9)- Е +2с^
whеrе с is thе еffесtivе diеlесtriс funсtion of thе individual two-laуег inсlusion in thе dipole
approximation and givеn bу thе rеlation
- €1Glp-2)+2в,
o = o.
-
, с,+€,(3/p-|)
A dеtail studу of sсattегing and absoгbing propегtiеs of suсh рaпiсlеs in thе linеar
appгoхimation (with rеspeсt to the еlеctric fiеld) whеn theге is no nonlineaг tегm in еr (l.l) is
gil,еn;n [l2]. iоmbiniщ (1 10) and (l.9) onе maу еasily show that thе еnhanсеment faсtoг
lt.q) anothe polarizabiiity сoеffiсiеnt (l'9) havе thе same dеnominator A (l.4). This mеans
that a сonsidеrablу inсreaiеs *hеn thе frеquеnсy al approaсhes to one of the frequеnсiеs (l.7).
At thе samе timе' thе absoгption of radiation by thе paгtiоlе inсrеasеs due to inсreasing in the
polaгization.
2. optiсat bistаbilitу in сoatеd sрhеrе partiсlе
Lеt us геwritе.}::"j;::T:iTjЬ:) in thе follоwing form
(1. l0)
(2 t)
а =zкe(})' b =
170
еL !егm ln
:з:еnds on
t: ihе loсal
fi.g a new
с-]tion foг
In a gеnегal сasе thе loсation ofroots ofеquation (2.l) on a соmplех planе dеpending on thе
сoеfТiсiеnts а,b,с is givеn in Тablе l
Таblе 1 Appеndiх
Rangе of parametегs
хз- а/+bх+c:0, Q:(HВf
+(G/^2<0,
Loсation of гoots on thе сomplех
plane
Н : -а2 / 3 + b, G:2(а /З)з - аb / 3 + c
in this сasе all roots ше геal
(l 8)
:: :.':ее real
.lg.: sесtioп'
Еrетanсе of
lз.::гiс fiеld
Етj : optical
в .s of thе
l .... inсidеnt
вi rсlusion
(1 e)
l : ..э dipole
аb-с,:'0,
с..'() , b-0
:l,1-С>U,
с:>О, b>a
,zb-i -, C,
9- ;1, b': Q 91
с iэ, b-4'
сlb.: .. i)
с<,.(] , b-,a oГ
с,.:.О. b_<0.
(l)
(2)
(з)
(4)
----*.-.-1--*
"'I
--"---*+--.-.-- ''
х. .сt'x., . bх- с=0, Q:в/3)
+ (G.,2)2>0,
II : -а, 13 + b, G:2(а/3), - аb /3 +с
in tЦs сasе onе гoot is rеаlаnd two roots arе сomplеx сoniugated
(1. l0)
ш '.'е linеar
t-. (1.1)is
.::-t faсtor
---s mrans
:.зs (l.7).
:.g in thе
(2 1)
аb-сr0,
с.-0 , b>0.
rlb-с',0,
с,'.0, b:,0.
$b-с <' c,
c',0, b>0 or
с>o, b.,0'
сth.с'.:, a,
с<-C, b:, 0 or
с'<,0' bK).
(s)
(6)
(7)
(8)
t
+l.*-fr---)
l.
n
о|
-_-.+-*+Н
.l
t"t I
In our сasе b>О, с<0 Therеfoге, dеpеnding:l: 'iun
of thе disсriminate
Q 2)
";-:";;-=ffi #fl [{i*,]i{#iж
ji*'*,Ы',l:'iJ.*Т.1;l
positivе roots Wе$lktxil;onе
to find thе borers.of IoB.
.".,..,i.ul analуsis bесausе of
Thеsеinеqualж:;жШЩ;l':":,Тtтк:т;:l.*..ъ.*'фiЦf ."i."i,io*.:l"-1'.::
tь. шgi.po*"'' ::,l: our paпiсular сase, .'J; ;""-'; ;;,:_":^..1.::".l-..,,'] l}ъ1; il:
ж":ilil";J;'"i/:':i:';+:r"t-1rl'.J,".ii}f ff :Т:',:y""lH:Ъi{"..Щi:t:
i*1,...ъТl-]}lll:|JJil;|+iLi*;"*l;#4l;i"1ry:l;.;;;;;;""Jiotь.
f;Нl!i,:,:Ттх,*ff?Ё'
i'ii" in,.*"r (x r, хz) Тhеse сondttlor
а1_.,J 3D' Qз)
f (х,).".,/(,')'
ше thе roots of the quadratiс еquation/(x)=0
ln our notations (2.3) takеs thе foгm
Rе(A. tq._J5lm(Ao/6),
. 81 \e'l e 4)f(х,).тv\-А',.,u},,.
х', = {-2Rе(Ao /;lтnEЪt.цo /6)], _3[lm(Ao l D)I- } l Э,
Ao = (яj ). + сi]|rio,(3 l 2 p _r) + 3 / p _ 1l + сio, 6 =1 + сip l 2 p -1).
Sinсе in our сasе el0>0 and,:]:,,^,}.,. is no IoB ri:т"TТi::.il"fi1il.:Е;
Rес2(o)>0 ть. ,..onJi""o"Ъ'n, "г
tz.+; dеteгminеs thе гange ot tnе
uoo".,\o*
wе again сomе baсk.,".:l:,::|i,J;:J::жL:l}]i:1"e:"1ij}:J''*'*n
o.n
"г
,' i' nеgligIьlу small In this situatton Inе lll)l wvr!v.-.-- (2 4
Ao6<0 prеsеnt it in thе following waу
Q 5
Using thе геsults oi thе first sесtion, wе maу
t, *,l:,lt, -,li,)(,, -@,') < 0 ' .^ -.\stitutеd with €roand
wherе thе frеquenсiеs Ji",u)"givеn bу formula (1.7) whеrе 61 lS SЦ0
(2 t
0't=+' eo =1/(3 lzP-1)
!"-
It is сlеar t|lat оJa)
".
<7!o)
"'
<aJo\
"t.
In addition n this сasr
1'72
22)
!rtl. геal
l
.t,,= -s\,,'j, :: = _Jf., id,
l,\
.'f(х'') -. (a),- i.(i. ).-. 027 ё'
Summing up thеsе геsillts, 1\е сan statе that
рaгtiсlеs r,"ith thе Kегг-iikе nonlinеаг diеiесil"iс соге
mеtal shеl] with nо dесaу (thе Dгudе t1'pе <]iеiесtпс
thе followlng frequеnсу bands'
(2 7)
thе IOB еmеrgеs in thе two.layer sрhеriсal
(diеlесtriс funсtion (1 l)) сovегеd with the
funсtion (l 2) lvith zеrо imаginary paгt) in
(2.e)
| 2a)
!с:-sе of
Е:].s on
в;е if a
l,...: linе
tп1 ]oints
E -'rt >
в: :c thе
('2з)
0 <сl<trl(o),' anсl ссi.,,.'. t.,,.-0l10],l (2 s)
Тhe IoB doеs not ехist aЬovе thе еxtегnаl aрpliеd fiеlds that eхсееd thе сritiсal valuе
Iпf]uеnсе of a smali imaginary рart оf t2 сan bе pегfoгmеd by usrng an ехpansiоn with
iеspесt tо thе small рaгamеtег and sllqhtll сhangеs the obtained гesults' тh"
"a,"
with finitе
.jеса.ying геquiгеs numегiсal mеthocs аnd ц,ill Ье done in thе nеxt sесtioп'
J. ]\iumеriсal саlсu]]t1оns of IoB in сoatеd sрhеriсal рartiсlе
ё--?,.+'-4-
|-==:::=:-..-'-
Y, i Ео i20.1 О 2 О.1 О,4 О'5 o,6 О'.7 ,
l. p:0 2, сm:i.5, ginЁl, Q=гоlбo=O 1' Г:0.01, e,19=J
1' lil iJ
.
-; ---
.n/''
Fig. 2. p-0 2' sm.-] 5. еjnЁl, Q=0.1' Г:0, €,16=5
4i
I
-l
JI
I
zl
\L.з ]
3: ' whеrе
Jъ-::е IoB Fig.
ь
I'
:.3glnary
(2 4)
(2 s)
.-t
(2 6)
{
.\
l\
/
. 1 i ..: l
|7З
Fig.3. p=0.2,6m=l.5, ainЁl, Q=0.1, Г=0.05, Еrtо:5.
n ^i )L.,t / ,'/
|o ,i эо. il эТ
.....;.. l.i i !,i l].
Fig. 4. p=O.3, tm=l,5, sinЁ1' С)=O.1, Г=0.01, Е,l0=5
Fig.5. p=0"17' Еrn=t.5, еinЁ1, С)=0.1, Г=O.01, Е110=5
1'7 4
:
. - , Эгl0-J'
l: l,ro:5
'fLv -.
i.6 rl,$ ,l.i iь
Fig. б. p=0 2, еm=l.9, еinЁl, o:0'1, Г=0.0l, t,to=S.
4. Еffeсtivе dielесtriс реrmittivity оf nonlinear сomposite
rvith two.layеr inсlusions
While studying thе pгoсеsses of inteгaсtion of еlесtromagnеtiс radiation and matпx
dispеrse systеms (I\,Ds) thе mеthod of effесtivе mеdium is widеly used. Usually, a
nonmagnеtiс МDS with disсretеly distributеd inclusions (the diеlесtгiс pегmittiviiyа;
еmbedded a continuous host matгix (thе diеlесtгiс pегmittivitУ €) is substitutеа wiiь a
сontinuous mеdia that dеsсгibеd by thе effеоtivе dieiесtriс peгmittiйty Е depending on Е,
а' , a dеnsitу numbеr of inсlusions rl, and thеiг statistiсal distгibution in thе host maйx. This
mеthod woгks espесially good in thе long wavе lеngth approхimation. Below we usе two thе
most rеliable appгoaсhеs foг сalсulation F.
At small density numbегs of inсlusions whеn theiг геlativе fraсtion .f = ! *i n is small,
J
,/<.l' thе Мaxwell.Garnеt (МG) appгoximation is usually usеd [9]
d_'^
=7Е_e-т-,'.=J -€+2ц (lз)
At higheг dеnsity numbегs of inсlusions thе Bruggеman approхimation is morе геlеvant Il0]
"F-E '. "-E-сf т -,ё+(1- f):__= =0 Qз)
Hеге д. is given by гelation (l 10). In this геlation wе have to inseгt €t = tto + 1|E'|, ane,
obtaining the loсal field (oг more prесisеtу z|Ё,|, ) fтom equation (2.l).
Now wе сonsidеr thе сasе of small amplitudеs of thе extегnal fiеld .Еo whеn
t- tz
xlE'| .. а'o . Еxpanding (1 l0) with rеspeсt to smallparamеtег' wе may obtain with thе samе
aссuraсy thе еffесtivе dieleоtгiс funсtion of thе inсlusion paгtiсlе
175
I - r2
ё =Еo+т|Е,| ,
(3.3)
wnеrr
Е^ -- с^o-,o1|
p ..т,'l+z9., - 90/ e_)': '-= (4.3)
o _ -2
€1o+в,(Зl p_1) ' r = хG;;'s1 ,-цL, \..-,
Whilе сalсulating thе nonlinеar еffесtiuе diеlссtгiс pегmittМф we follow thе sсhemе suggestеd
in Il l]. In this papeг, t'il;;*;l;;; й' . 'y'..',ii""'i'.ьЁ
of N nonlinеar сomponеnts йth
diеlесtriс funсtions Ei =бoi +r'|Е,| ('=l,...,}0 and геspeсtive fraсtions l thе еffесtivе
nonlinеaг diеlесtriс funсtion is givеn by the геlation
N ry t- 12
E =io+'jif 4l4l lЕ.l '
i=l Ji
^хvlhеrе p, = ?Eo and Еo is given by thе gеnеralizеd formulas (1.3) oг (2.3)
Uo
'О
Eo-t. -*,
€,o-Е^
E^+,ц- Ll, Е,o+2€^,
Ч r 6,o-6o _0
ft"' с,o +2vo
In thе сasе of gеnеralizеd МG and g.в8":1n formulas (6.3) and (7.3) we obtain
(s.3)
(6 3)
(7.3)
(8 з)
(e.3)
10.3)
9.f ,s ,
tr €'o-€,,z
;,o+2ц)1с,o + 2с,), (| _ZI,
=' - 'f'EoГ, =- -. . Е.l 'f,Е,o(t,o +2vo), нG:;Ef
Sеtting N=1 in thesе foгmulas wе gеt
9.fс2Е-
" -
1to*2t, jG"_€)l,,
t, - Eor
', _
ёo,f + BQ_ f)r.
^ ,€n +Zdo,,
wheгep-_\^-'; l
b Й a Loo
Foгsmalldensitуnumbеrsofinсlusions,/<<l,inthеlinearappгoximationwithrеspесttothis
;;ШЖ-;;: bJ ;йHH,е thе samе rеsult
(l 1'3)
,J"j"; $=Ё.''.n
for thе еffесtivе nonlinеar diеlесtгiс permittivity
(12.з)
€=€o*I|Lo|'
whеrе
1'76
.!,.1
! 3.jgеstеd
Els йth
l ::есtivе
(s 3)
э3)
, 7.3)
,.8 3)
(e 3)
10.3)
ts:есt to this
(l 1.3)
г.( эеrmittivitу
(l2.з)
vo = с,f1*з1 !э:З,-J, ёo = €z
ao т.ёm
а'o(3/p_2)+2r,
o,o + or{31 p -1) '
i=z( ! )'.1 ' l'. gf(ttP-t)":
.2+€a/€^
!2+сo/с.l [с'o +с,(3/ p_I)]
( 13.3 )
(14 3)T
Foгmulas (l2 3) and (1З 3) еnablеs us to find thе еffесtivе nonlineaг diеlесtгiс peгmittivity of
thе соmr.о.itc !!cing thе known diеlесtriс pегmittivitiеs of thе inсlusion сoге е1' thе mеtal shell
Е2, and rhе maтгiх mateгial е' pгovidеd that x|Ё,|. ..
",o
and/<<1
Now wе notе that thе ехpгеssionfor I maу bе wгittеn in anothег foгm bу using the
геsults of Sесtiоп 1:
--х .f(1-p
.7
29 сiс ]|с," p + е,(3 - p)|,
|6p"|(а. - e,-)(.с, _ а,- )],l(g, - с,-)(сz - €,)|,
Ггom this геlatiоn onr сan sее that at frеquenсiеs of thе inсident elесtromagnеtic гadiation
сiОsе to thе fтequenсiеs оf suгfaсе plasrnons and pгovidеd that thе imaginary paп оf о2 is
соmparativеlу small, a сonsiderablе еnhanсеmеnt оf thе еffесtivе Kегr Ъoеffiсient j maу
оссur. Тhе numегiсa] evaluations show that foг inсlusiоns with silvег сovегing a ratto | /7 maу
bе of thе огdег of 1 05- l 06-
Сonсlusions
Тhе pегspeсtivеs of usagе of thе noniinеar mеtal сornрositеs (NМC) in sсiеnсе and
еnginеегinе havе bееn disсussеd paгtly in [1-5]. In thеsе papегs' somе typеs of non1inеaг
diеlесtriсs wегe namеd as thе most pгomising foг a fuпhег studу.
Hеге ц,е would likе to stop on somе thеoгеtiсal сonсlusions that follow from thе
rеpoпеd study. Fгom thе геsults obtainеd in Sесtion 1 wе сan sее that аn enhanсеmеnt of thе
loсаl fiеid in МDS with two.lavеr inсlusions is possiblе only pгovidеd that a геal paгt of thе
diеiесtгiс funсtion oГa сovегing must Ье nеgativе. For thе diеlесtгiс functions of Drudе's tуpe
it takеs plaсе at fтеquеnсiеs О 4 0, l J; It is known that in this гangе of fтеquenсiеs thе
wеaklil dесaying surГaсe modеs ехist in mеtals. Thегеfoге, thе mеtal оovегing of thе nonlinеаг
diеlесtriс makеs еasiеr an еnhanсеmеnt of the loсal fiеld to thе геquiгеd lеvеl,
onе morе impoгtant faсt is also woпh noting. Foг thе following NМC struсtuгеs thе
]oсal fiеld may bе found by solving a сubiс еquation: a twо-laуеr planе struсtuге, a sphеге of
nопlineaг diеlесtгiс in a mеtal hosr matгiх and a two-layeг еllipsoid of a nonlinеaг diеlесtriс
lvith a mеtal сovегing ргor.idеd that thе extегnal elесtгiс fiеld is parallеl to onе of its aхis' It is
tуpiсal that foг thе Кегг tуре of nonlineaг diеleсtriс thе сoеfiiсiеnt D>O and thе сoеffiсiеnt с<0.
Ilоwеvеr, thе oгdег of the сoгrеsponding еquatiоn is еqual to five for a sphегiсal anisotгoрiс
diеlесtriс inсlusion with a mеtal сoгe. Foг the two.layег еllipsoid that is similaг to ouг sphегe iп
thе struсtuге at aгbitгary oгiеntation of thе ехtегnal еlесtгiс fiеld with rеspесt to thе еllipsoid
eхеs thе огdеr of this equation bесomеs sеvеn' Thеse сhangеs in thе ordеr of еquations maу
lеаd tо thе гiсhег piсtuге of thе IoB in thesе systеms. Moreovег, taking into.aосount thЬ
damping of еleсtгomagnеtiс radiation may сonsidегably сhange the сonditions of the IoB and a
magnitudе of thе еnhanсеmеnt сoеftiсiеnt'
t17
^.*"i#1iinT,.,"j.o""wlеdgе trnanсial supрoгt from thе National Sсiеnсе Foundation
thгough thе Faсultу вu,iу с",!., oеvеlopm#"iёдкввкl Awагd ЕС5-9624486 and an
Еastеi Еurоpе Pгogram Supplеmеnt.
Refеrеnсеs r rrr r: z^t.i., Qrotinnaru nrооегtiеs of dispеrsivе
1СсSung,CМ.Browdеn,J.WHausandw.KChiu'Stationarypropегtiеsofdispеrsivе
ооtiсal bistabilitу i".ё"ёi)7pьуs-к." B . 1984 - V 30, N 4. . p.1s73.1882.
z d Jungk, optiсal Bistabilitу in Сompositе й.oiu zz Phуs Stat.Sоl' (b) - 1988,-v.146. -
P 335.-140 .-:-',^l1о D Inоttvя аnd C'М, Bowdеn' Nonlinеar.optiсal pгopепres^of.
:"rJJiiJ;,Т:Iнш:}'}; ll*#''Ё,",i,*,do]*;;;:-,-;;;;-. iss, ьr 4 .P 1g'7-
.c07 \Il aJ.,,о P Tnоrtvа and М'H' Biгnboin lntrinsiс optiсal bistability for
4 N Kalуaшwalla, J'W' Haus' R' Inguva, at
сoatеd sphегoidal p"ni.i.' 7u Phуs й.еv. A - 1990' - v'42,N 9. . P.5б13-562l.
5 Y Q Li and С.С. 5,"*. *Ъыi"Ju,-opti.ut p,opй.' of sеmiсоnduсtoг сompositе matеrials
., ьl!Ъ;':;ъ*'oxl]1. ";=,,-}fr Ё'l'1i;1il9,:9t]i approximation for a familу of
nопiinеaг .o*oo,,i.,'iiP;y" R.i B. :
1-992 -v 46, N |1 -P.1|89-,7192.
7.ohadLеч-,YoadYagil,and.Davidl-в.,g,nunFiеld.induсеdtuningofthеoptiсal
рroреrtiеs of nonlinеariomрositеs n.u,,.,-onin"" ,t l Apрl Phуs. _199з. . v.76, N 3' -
P 143 1- i4з5. ^ -.'^.,.^- ^f T iоЬt Ьv Smаll Paгtiсlеs, -Willeу,
8.S,F.Boгеn,DR.Huffrnan'AbsогрtionandSсattеringоfLightbуSmallPaгtiсlеs,-Willе
, ;а l?ll;Jl,j.;jfi Ё с9roy; in mеtals glassеs and in mеtalliс films // Phуs. Тranс. R
-
so. -1904 - v 120з - P.з85-420
10' Brugегman о д.Ь" в.,*h.ung vеrsсhiеdеnеr phуsikalisсhег Konstantеn von hеtеrogеnеn
Substanzеn ll аnn. вьуs.i9з5-- Y
'?1
.P 6з6_619
1 t' o, Stтoud unа vв. wЪЪd. Dесoupling^upp,o*i'*ion fопhе nonlinеaг-optiсal rеsponsе of
соmрositе n,,"аio7 l.'6pi sЪ. д,.,,,
'.,
j?dd - v 6, N 4 -P .718.186.
12 L G Gгесhko, v'Ы иi."сovskii, V.V_."йo'i"й'й V.N Volokh // Adsoгptiоn Sсiеnсе
',
ttъ*.";нJi'?.;;}i!*.'l' *"ifiil],'iеlесtгiс funсtion оf aggrевatеs-9|9mall
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178
|
| id | oai:ojs.pkp.sfu.ca:article-56 |
| institution | Surface |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-24T17:44:39Z |
| publishDate | 2001 |
| publisher | Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | surfacezbircomua/3a/f5b4c8d30e5253ffc063f84b8e87c73a.pdf |
| spelling | oai:ojs.pkp.sfu.ca:article-562018-11-27T09:42:39Z On the problem of optical bistability of nonlinear composites with coated inclusions On the problem of optical bistability of nonlinear composites with coated inclusions On the problem of optical bistability of nonlinear composites with coated inclusions Grechko, L. G. Davidova, O. A. Mal'nev, V. N. Whites, K. W. The electrodynamical properties of the nonlinear metal composites are intensively studied in many papers [1-7]. One of the most important property of such systems is the abnormal enhancement of the nonlinear optical respond in the composites containing small inclusions (compared to the wavelength of radiation) of a nonlinear dielectric covered by the metal shell embedded in a dielectric host matrix [2-6]. The surface plasmons in the metal shell may be tuned in resonance with the external electromagnetic field and produce a considerable increase in the local field in the core of inclusion to make the nonlinear part of its dielectric permittivity to be important. As a result, the connection between the applied and the local field in the core becomes nonlinear and in some diapason of applied electric fields even ambiguous. It happens to be that one value of the applied field corresponds to a few values of the local fields and polarization of the inclusions that in its turns leads to instability in the composite optical properties. This phenomenon is called the intrinsic optical bistability (IOB). In this paper we calculate the dielectric permittivity and polarizability of a separate inclusion_and analyze the parameters of the IOB. Further, we consider the dielectric function of The electrodynamical properties of the nonlinear metal composites are intensively studied in many papers [1-7]. One of the most important property of such systems is the abnormal enhancement of the nonlinear optical respond in the composites containing small inclusions (compared to the wavelength of radiation) of a nonlinear dielectric covered by the metal shell embedded in a dielectric host matrix [2-6]. The surface plasmons in the metal shell may be tuned in resonance with the external electromagnetic field and produce a considerable increase in the local field in the core of inclusion to make the nonlinear part of its dielectric permittivity to be important. As a result, the connection between the applied and the local field in the core becomes nonlinear and in some diapason of applied electric fields even ambiguous. It happens to be that one value of the applied field corresponds to a few values of the local fields and polarization of the inclusions that in its turns leads to instability in the composite optical properties. This phenomenon is called the intrinsic optical bistability (IOB). In this paper we calculate the dielectric permittivity and polarizability of a separate inclusion_and analyze the parameters of the IOB. Further, we consider the dielectric function of The electrodynamical properties of the nonlinear metal composites are intensively studied in many papers [1-7]. One of the most important property of such systems is the abnormal enhancement of the nonlinear optical respond in the composites containing small inclusions (compared to the wavelength of radiation) of a nonlinear dielectric covered by the metal shell embedded in a dielectric host matrix [2-6]. The surface plasmons in the metal shell may be tuned in resonance with the external electromagnetic field and produce a considerable increase in the local field in the core of inclusion to make the nonlinear part of its dielectric permittivity to be important. As a result, the connection between the applied and the local field in the core becomes nonlinear and in some diapason of applied electric fields even ambiguous. It happens to be that one value of the applied field corresponds to a few values of the local fields and polarization of the inclusions that in its turns leads to instability in the composite optical properties. This phenomenon is called the intrinsic optical bistability (IOB). In this paper we calculate the dielectric permittivity and polarizability of a separate inclusion_and analyze the parameters of the IOB. Further, we consider the dielectric function of Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine 2001-06-10 Article Article application/pdf https://surfacezbir.com.ua/index.php/surface/article/view/56 Surface; No. 4-6 (2001): Chemistry, Physics and Technology of Surface; 168-178 Поверхность; № 4-6 (2001): Химия, физика и технология поверхности; 168-178 Поверхня; № 4-6 (2001): Хімія, фізика та технологія поверхні; 168-178 3154-8091 3154-8083 en https://surfacezbir.com.ua/index.php/surface/article/view/56/55 Авторське право (c) 2001 L.G. Grechko, O.A. Davidova, V.N. Mal’nev, K.W. Whites |
| spellingShingle | Grechko, L. G. Davidova, O. A. Mal'nev, V. N. Whites, K. W. On the problem of optical bistability of nonlinear composites with coated inclusions |
| title | On the problem of optical bistability of nonlinear composites with coated inclusions |
| title_alt | On the problem of optical bistability of nonlinear composites with coated inclusions On the problem of optical bistability of nonlinear composites with coated inclusions |
| title_full | On the problem of optical bistability of nonlinear composites with coated inclusions |
| title_fullStr | On the problem of optical bistability of nonlinear composites with coated inclusions |
| title_full_unstemmed | On the problem of optical bistability of nonlinear composites with coated inclusions |
| title_short | On the problem of optical bistability of nonlinear composites with coated inclusions |
| title_sort | on the problem of optical bistability of nonlinear composites with coated inclusions |
| url | https://surfacezbir.com.ua/index.php/surface/article/view/56 |
| work_keys_str_mv | AT grechkolg ontheproblemofopticalbistabilityofnonlinearcompositeswithcoatedinclusions AT davidovaoa ontheproblemofopticalbistabilityofnonlinearcompositeswithcoatedinclusions AT malnevvn ontheproblemofopticalbistabilityofnonlinearcompositeswithcoatedinclusions AT whiteskw ontheproblemofopticalbistabilityofnonlinearcompositeswithcoatedinclusions |