"ffi8ii,tH["i:?tls3]`s3#,5:it",,#j$h`i:#sl`dft!!`:Hi13:1.,5gt",,9i,tJ3".

"ffi8ii,tH["i:?tls3]`s3#,5:it",,#j$h`i:#sl`dft!!`:Hi13:1.,5gt",,9i,tJ3".

How the stiffness of the optical trap depends on the
proximity of the dielectric interface
Potr Jjkl". trIojrlir Ser.v,.I Lr-o:lavLislar,ur,L far rl Zelrarrek,,
olnstirrlte of Scieutific
InstmrleDts of Acaclerrryof Scielces of Czech Republic.
T i t - l . r ' o t , ^ l r k i| 1 . . L , l : o t B | r r o .C z , , L R e pr , , l i ,
'lustrirlte of Phrsj.al
Eugileerirr3.Fzrcthyof \Ietharical Ergirieering.ljrno U versitv of
Tcchnoiogl. Teclrrrickri2. 6lij 69 Brno. CzccL R.cplirli,
ABSTRACT
\\'her d pfrt)e coufirfll nr ,r single focrNecllaser Lrcanrrrpp|otrcho
the s,d:Lce. ir is gcfturg lrore iljiuexcecl bv
r|e
rctroreflecre(l
be.n, TLis berlr interreres
witr,th€ nx,i.tentol-" .nd l,*"r
,,1."
- :;,;;J
\ ' l i i c h S l i 3 I l l h ' n x x l u l i r t e s i l I e n I C i l l . l t l J e d n ] ' $ . e s t L l d i e i t e x p c I n n c l I t h ' l l o w t h i".,ar".
S P i i e n o t L c l l nisri;
nlljle[cestLe
opri(trl tttrP prop.dies if s\\: is crente.lrrsirs su n..s ot r\o dift.r€ni reH.,ti'it;,,".
\ii:usett polvsrvrele probes
(f(lixlll('tcIs.j90!uiD(|82()lillL.tln(k'.Itlleirl)osjtn'ls
tlcur.il rxrtio fo sft rLc a]:jnl rfap stiflircss along optical axis.
Keywords:
ot)tii.,rl t$1\-rcfs. pnrri(.1. tr&ki
g. Bfolvriid! ubtn
. opri(.^t irarr sri$n$s
I. INTRODUCTION
oPii(nlthl)Sllascr]orIasir(l('lbc||s.(llit{'|buIu(so.a]]{!lsiltg]{.lX'!llipssBT|)
']{\ll|]nll5}i|[l],,so|s(i('lltifiC1o|k..!tI('h,!scoll(,nllllcoIli*;.:
r'f r"i!rrr s
l l r t r r ' r o r t r r h ' s s . t L t i , , lt o k r ' o s 'o l ) t i ( d l t r . p
Irrperrics nn(l |o1! iL(\ rire,tfi.ctirll), elipernrrerr
rrc(li r! *'
lpc|tlu1'llliu1,sCopeol)i(Jn1''Hov'v.|.
\'rt.l-glrssiIltel1't!'(,'tllcsll|l!'oftlicol)tn]t|'|]'l|l||i\P|''|{ltj\,|'''ftt'n11l^
of hi.rLl! (,ti(rius
rk,irrtrrlacc.'rl
t , r f t k h , r p l r r ) i (l x s , r x i , r l 1 1
i r r ' l x . , 1 1 s ( f i | l \llr L a r r g $ i r rr r l r a t n r r r r )
i , : ( r ) + ^ r o : l r )+ h . : { i ) : - j t ) .
(i)
(listrr(r,
$l'.k,:is
tlt'nno(l!1|tr1ui..rllolD]llrirtLl|.'h:istll('trrl)sti1h.Ss...0:{jn|,liSsbk's'
()t|d|ir|SfiliIl|lIi1lulo|r'i:coSii\,'rl(],-l(/]is11rltrratll1llutl{llllc
,,, . ,,,.,'i..ti*
l ' . . i: l l - r , r 1 l i . L k( l i s i r ' I ) . ,r . ' l i l , . l n ! , , , . 1j t : t ) i l t ; l t . : ,
1.f.\: +l20ill
i| l ti)2
*"ffi8ii,tH["i:?tls3]'s3#,5:it",,#j$h'i:#sl'dft!!':Hi13:1.,5gt",
Proc.ol SPIEVoI.595859582pl
ouritte(]. Usi4 Fourier irinsfofi s it is possibleto oxpresspow.r specirnLdeusitJ-(PSD) of trappnl plrrticle
positionh Lorentzianforur:
n,iti'l
rsD =:----:-:
t'+J'
t.tr
is the diFusioD coefficierr aod f. = x,/(.2rto) is .i corner freq,reucv of the Lorelrtzi.ur.
wlrefe D = lsrho
UnfortLrnarelLy lhc expernrentalll, obtdnled PSD di1lels froln th€ siurple forrr (2) becnuse of ihe lbuowirs
1. The re.l drl. urreacquired during liDite time iDlervzrlar.l so it is better ro LNenliNed Lorentzitrrr.
I
TIie probe is not trtrpped ir free sp:rce. The vi.iriitl' of tle nrlerf^c. nrltueDcesthe pirtirle Dol-ement. It
is possiblt io erpress rle irrfluelce witl Ftlrdu s fornroh"
r---r 3 i#
#
where R is radirN of the sphericil pariicle l1ud I is distance beteeeu cenier of ihe p?rrticleind rhe plane
J. Thc pfolic is nrovirg i1r nrconrtlressil)lcfluid nr Io$' Revuold's Duurber sysienr. ThlN. the friciion for(es
dctirs on tltc probe ^re iufluen.ed bv pnst Dro\rDrert of the l)robe throLrglr th€ flLri(l iurd tlre oscillitions
further nro.lii Stokcs @ettifit'rt:
-1r):,0(r+
, ,r+r ' # )
(1)
whc( 0 is liequcrc]{Io
ns distdcc fronr th. sphcrc.
Tlic rbore
poirits Ndc
Put tulicthcf
dnd so .rll.d
phvsicdl poNcr speftflnr
$rl.s fol
ulntr'(l.''
,,/(2'r')Rf./,,
R/t)
P,,,t,,,(f:
(5)
(J,+ tru,-.i-,- J, /1,,,)2
+ ltR\'-,/-,,))
'!lLe|{)
, r r _ i 1 Tl , ( ; " , l ; ) ^ ( i . t r )
n.
arLrl
1,,: r/\"n)
fn - 1/lram)
:
',,,...,
''
(1,,m,)
"rLr*f; (_7,,rr)",
i s f r c r l u e n f f o r v i , i ( l i l , c u c r r d t n ' , ( l ( r , t l Ld - = , q n , I ' l , r i . lw i r l i k i l ) t r t l r i c \ i s f o s i r ! z d , ! l
is ftcqrk\r'r' of nx ftiJ r.lNntn'i for plrti( le 'virh ui.r-ss,,.
1'Scnii(|on.ll(tor(k't.ctoflctsnsnl(xv'Prssfilkt.b..^lIs.tlloik'i.ciid]s
Tlle chrh(tcristi.s of ()PD is sruL of snr^lL (oustdt .r('rn'r!r:r rt Pf{.1(.rLfirLo
nntrredintc resporsc tud
Lomrtziru (tresextirg the (lel,rtlrl rcsl)onse.
f("/)
1
.rl{li!rt'!
. . l n n r i . ) :-i+ltArq*t
,
,",,(/)"
(.t
T l t r , l r r e s , t t , l / ' ' ' ' . , ' , , 1 / l : l i , ' " , r " , , , l i L r h t r t\ rl o r df i t t i r Lrsh cP S D .T l r . f i r t i lr i t t l , , Nu s . sf b , r rf t i , , s
I i . , r .. ' r e | \ : J . . r . r '
AIiu tl!'firirl
( l
. t | f' r r 1 ,r . '
tit is lx!1brtrul rrrl rll lrrrrux't.h nro ol)tann\I. tho SBT sriftu$s is fnl{'lllirr(l ILor' (orr('r
t;. :21ti l, .
NLem -/ is Siok's (oofici(jir couectc(l to surlh(r. !i(niirlin.l
(i)
osiillrtiris motidr iu n!ou!rcssil)le
Proc.ofSPlE Vol 5958 59582P-2
1tti(l.
2. METHODS
2.1. Experimental set-up
TV screen
computer
IR fiker
l2-bir.+/- lV
tube lens
dichroic
mirror
laser
filter
rerescopes
D/A converter
l8-bit.+/- 5V
objecti!e
PZTs
SAMPLECHAMBER
DETAIL
lns apenurc
r sllp
spacers+probes
-quadrant
microscope
slide
Figure 1. Opli.al IN..zets .onrl)in.d sill
p-p6cessine
Photodiode
;nit
iIumination
l)nrtile posiridr (l.te(nr'. 5.. lhr rext fir (t.raik{l ikscripli(h.
ThcaPPliriLtj|s(sccFiA'1)isbllilt,ti|ol|1(ll|l)ri8htltrI|,:ito|vtun)s.(,lx,(()lvlIlPls
s o l n l * n n ' l n s e f ( N ( l : \ - \ O r . . \ = 1 . 0 5 : jt r n r . S p ( , d l n P l r y s n . s2 0 - \ \ ' - t 0 i c ) s e m , s . s x s o t l r c . o f . d t i a i n n r . T l L c
t|lppnigbonuLplss.st1uollglLl)nn(l.PNsfiln'|tor('n()\.cLisl]tofi0illl8ldse.(llj.k'sl
tll'P.Se.on(lh'.iti.])ossil)l.nlsctt]t'i|rl)l)osiri1'IitlI'
|ig'l)isld'lt1\t|x'ii'ccD.o!('|51il,u{1|n)ttol]lsll''
l.1.0l/ll'P(nvs(i('|n1's.P(n11{..i|)'Us.(lpiIti('lcsPt1x's
.rtl(l82()Ilul.D|lk.sCnlltifi.)^||s{,l1ltc(li!llo$
s l : r s s( r e f . f t i v i t ! ] l : l ) . 1 7 i l w . ' h , r) o r d : t l d s sc o n k r l l r I s l s t e x r o f r l n , l ( , c i r i l(, . r s ( R : 1 ; J Z i r i N { t ( . r ) . \ \ - h o l t ,
sl'tiI)lt.]'u'u'|i!^slttlc1l(tlto|lanol)oSitioDillgstd3.oq||il)I)e(|!'itll.npifiti('s.nSors{PI517'ic'Phlsik
hrstrrr!'ltc.
tlnnr3| tlllx li\rs to eirhcf CCD fiinri,|l of r\{!x\f.
P o s i i n , r o f t l l e t n p l ) { l p r o l x N r s ( k . t c c n r l I s i l s r q r ( L r n r r ) t r o i o ( l n r l c( E C ] S G . U V l l t ) B ( ) - t ) . T | c
(li tc.ior ras phcf(| to tI
prttcrn f'cnt(d
eqlul&)()'5nJ,(lli(\1'lrrr1rrlrlllr'col4lrlliso|
( ) P D o l r t p I t \ r s . o l k \ t . ( l l ^ l l i l ] h - s t ) ( q l( h t a r c ( t r t i s i i i ( n (r r r ( t
(horr('rrr,r(I1
i 8bir.+5\')
{j.0).
l C ' o r p r r r c rB o D r l s . p C I D A S t 0 t 0 , / 1 2 . 1 l t ) i t .
u s n r s C A \ i l l r c f n t r l . T h . ' ( o u i r o l s o l i N r f . w i r s\ r i t n !
Proc ofSPlE vol 5956 59582P-3
i'l C/C++ (tlicrsoti \,isrrrtc,
2.2. lvleasurement procedure
The poh-mer prob. was tfipped in the snNple sptire,rnd pushed to$'ar(ls the botton slide, so thdi the trzrppirrs
bennr wiist e'trs fbDred se!€rnl rnicronreie|s irside the sLicle.The s?rmr)leNas th€r nlo!€d up in .10 nm steps.
In ench ponit o{ nrensuremert iliefe sils n shori delnr' (10 ms) io let tlLe probe settle ir equiLibriunr posiiion.
AlteN\ruds. 50000 tl\iaL probe positions r!€re fecofcledrlitlr stm\tnrg fate 60 060 Hz iDio sepafate frle. Acquisi
tioD rrnd pfocessirg of one position took dboLrt 5 secolds. EaclL serics consisteclof 100 poiDts coveritrg zlr trxi,rl
regio of 16 pu1. Since ihe lvhole ure:rs renrent lNted hnlf hour. the sinrple had io be prepdred carefull] to
prevelt dust pdrticles or zrrother dirt f.o r enteriDs the optical tfnp an(l clistrubing the posirion sigDtil. \\a rNed
distilled srief fufiher clenned by srrirge filter u t (PVDF Dienrbrar1(..lliLlex. Uillipore) with 0.22 rlm pore
2.3. Data processing
Rawpowerspecttum
denslly
g
E
100
Timata.u.l
3000
2500
g
2000
3 rsoo
E
i000
500
0
F i g u r e 2 . E x a n , d r o f ( 1 . , 1p. r o . . r s i , , s .
h r o s r s d p p t r f tr t l r
Tlrr' (l.rtrt llr'rc' thcr pro.ess.cl b\ r \IrtLrb co(l( rirrl{' b! B.rs Sof.nsfD s grorrp.ri \\'r' rrulihe(l it to flr
r r x k r l l . t l , t b R 1 . l ( t L . o r i s i r i . r lv . L s n D s o k f ( l n r R r i J ) n n ( l t o e r n b l ( . b r t c h - t ) r o c e s s i r o
r gf t L e r r l c s c r i s o f
lr(,rsjrrfulfut.
ott1('Iesl1lt1telollsqllsillililL|on(liti(,NIi)Ith.filal1it'T1l.nois.$'ast1!ll
s i t h l J . ' ) (l )l a t t rl ) o i u r si I
tit|oxlilt|(l),FiiiilIg|lll(tn,is(5)lnd((')t{,tllcP(])[
tllitisirg. irLch(lql !1.tir lnrnD no(tifiQtnnrs ,rrl(l .ousnlof{l th{'()PD
Proc.olSP|E Vol.5958 59582P-4
low-t)nss6ltoriug.'l
$'hcn the ifitp stiftiiess ifkrrorv!. wo cdn estinate the.teiectof cnlibrtriion corstdDt nt ilis positioD. T[e ListogrtrD of pnriicle positnnb is Gdr$siilu $'hich nxliclres tlnr the rrtrp botronr
is pnrabolic (see Fig. 2d). It is theLelbre possible to estirute
the tfap stifiiress fmu equip.rtitioDnl theofenr:
x10-
g €
=-s
.9
1, _
1 ..,.
:
tkBr
tE^\.;)
g 4
Beamwaist- bonomslidedislance[]rml
F i s u r e 3 . r l , t , , . t, ' r o r . r l . ^ . . , . . ti{. i , . . ^ . t h . i r , ,
,J. ol prrli(1. positiotrd.rc(tor
r])
ehere ^" is the norLcalil)r.rted trap stifiiess (givo, iu
[NnrV
ntrd {.;) is rcot-nienn-sqlnte (RUS) of probe l)ositio! fiLrdrlrltxris. (:i is tlic non-crllibritect sigrul frottl the QPD given ir
[\-J). The cdlibratiorr constnut ai ihe curreur trtrp posiriou: is
obtdnied fron tLe conrpdrison oftrdp sti$i!,sses obrlnxal froxr
spectrdl dldll1is atld.quipirtitiolJ
rh€orenr:
(e)
trrfi)rtuu,rt( l)' \e obsor!1
is.loseItIllrll1pDtothesll|Il|e'Thc|ctisolisilLattllePotoutiilpro1il0oftlleirdPl()S()si|sl1nnr)Ii'nlhlK'
ur1. tlloredll'f. the lrhre of,l. slishth chnnges,isthe pfol)e rppro,tcl)esthc irrterfnce. t\'c nssurt sDroorncndDge
of(tlLih|l|tn'iconstal)td|(1soillsi!'n(lofllr'asll|d,Jji!o|ls..litselthpol
ifour prolx positirnrs 1 /rDr rln\1' rLe srur)le- In FiA. :l ttr('rt is nn.\arnple of tlie detectof c:itibfliion for r
sPI!.|oo|i12{)nulindiiUl(tcr'Tllc.lotssliowlLeisu|c(lvll'sofJ..soli(lliueistltepolvouli?llc
Th.valll('so|,j..irrcrcrrsc<Itr;rcvcl's|h.i|)rai|l)ositi('llliSiog|lxls'Tll|Sp|ca(lo|llctlldtilgPlobcs
{.N .stiurnt(ri filxn t|o rlif('llr(t of (tu{nrils ol c^libriLirrl probcs positii,is
l t 1: r 1 1 r i
r11.2i
Thc(Ilhntit\'/l/isilflI|||elirelr{\ltot$'.tInplLciglit...
b ! t i r i r l u l l ) ir i ( l c .
3. EXPERINTENTAL
RESULTS
D . r s c <rlI o r r r
((|illlx'tc|Ii2(Jxn''Ti'oi!P(s(Jl|cH.(titigsllrtri.s\1'n't1'stc(ltll.lIbstlicqlxrltlv
Rlsults trrr pr.sfnt.,l il rlul\'srrt)lis:
. R.1)r'(scntrti(rr
tqnilNtt]!'sll|fl...itti)li(^istll|ux^'1'l!utoftll|sta;cullsotI('sl(,Pcoft|f|st
i1, 1 Tliis all(^!s rln (liil)htnnr i,l rlL. (htr. (S(r tlf crrss.si| f. s. Fijl. 1).
.Fnl:l1thpsti1ti(sstll('uill|\1tnlilll\is'1.ll.nln\ilIltllsti11
l ) o t t o l r i l l t ( ! f e i ! ( l s o b o i h t 1 ] . s ( t r t ( ' r i r L rui -( l g f r i l i i r r i t b r . c s p r N h t h e p f o t x . U n i I s r t L e s l i ( l e .
. Tml) L.ijthi .rk Ilnt(l
s t i l l u c s su ( l t h ( ' I l r i i r n r l h f i g l r r ( , f r l L . r f a p \ r n s i l l r L . o f ( t t ! o f l l i t s o f M n d , r ( r , { s .
If t|c pnrii(h is scxsiti
Proc ofsPtE vot 595859582P5
(t0)
E
16
E
2
s30
a.2
E,o
3zo
g 02
i 1 5
.9
2
- 4 - 2 0
Boamwaist- bonomslidedislance
[Lm]
5
Fisure 4. Sph.r€ didnteter is 600 nn,- The boftonr slide is 6n ordinarv Eldssrith R:0..1% refleiivit!.
3
0.3
z
25
2
p
5
E
6
4
2
4
Beamwa st - botromsidedisrance
bml
0 n r . 1 h . L ) o r ns l n k .i s d c o a l . ( ls l t r s sw i l h / l :
F i g u r e 5 . S p l kr . d i n n r . r . r i r ( i 1 ) n
r:17n' ccis rrr
E
3 03
!
_306
7,0
I
P ^.
2
E
3
o
4
3
0
Beamwa sl boltoms'dedsrance[um]
F i s u r e 6 . S p h . e , d i r n t r , t { fi s 8 ! { ) n , , , l h . l { t l o t r ,
sli(k is.n orlirdN
Proc.oi SPIE Vol 5958 59582P.6
g l r l s sN i l h / l = 0 . 1 7 k l l { r ' l i Y i t v
F
; o.3s
E ".
3 025
E
e
i"
I
E,
€ 0,
F 005
- 2
4 , 3
1 0
Beamwai$ - boiom stidedtsrancelrml
Fisure 7. Sph.re (tntrr.ier is 820 rrr. Th. boik,n stide is r conle.l
lit:Lsss.irtr n:
t.l7 .eflccrivitv.
.
Tllf positn'r o I ihe sphef. i! s \\- in fll(rxr\s .lso rL.
r)ropcrtiesof th. rr.p. The t fap stifness i, sidble.. poirr
(nrfle\idi
of s\1.
of t|e.j,urpius strnci.re
l'ft grrrprLof riual figrues) is ;Uch high ihdn stitfless iu insi.rbre
po'''t('ni'lilnd/nlnxillLrofjunu)itLgStructu|c)'Th.stifIresSclianB.esllt
tllcjlIIlu)so|thospllerl'.b|lttllova|esItiftc(lbvlLr]fofthe'lunp]l.iglli'TlLefIlcilIntnnlsoft
'u. higlx r itr the c.se of s(''siri* spLerc sizes ((o'rde
Figs. r nud 6). The cni\.r of s\\' is rrurctrhigier rbove
thcmf(ti\1'sllr1llce.dSii$'tLsdticil)lt(\l'.{nilLt.r.siilLI]lnl|Iex
ralrt's rrbo'r' rrlT reflccrri'e slNs ^re j.sr slislrrlv lLilrhcrfoup,if(,.I tu rh.' poaks dbove
tlLe orrli*rrv glss.
ItCiu|besu.tll{tthothPhoi3lltclull8.sfonl2()nlj}5lllllnlt.(1se
(,ri]nlr|!gllls5(!t|igIltghI'lonFis'l)'whi]('tllesr!!,pnij.b|ltl
(f L('ight 15 n)!5 nnr (Fis 5) so. tIe l('ight ofth. t.np abovc
rh. reflecri\r sldss is l$sef rul
re t.rp rr(,ight
thethplioiglllS(u(nro|l(ss1()Iun)'TIl('r!crr3|1('iqlltjsstil1l('ss||{|!^1.
4. CONCLUSIONS
ixtod;lCcof||fe.ti!itfn:{),17c|('^k'snsllu(|ir
rvit1'noll('si!1!|dliill(xli's'Tl{'|1]fu.r.sisti$I1'sscS
st,uxlillgQt1,.Tllis|fittisf||rllluc!1trlx1\l''li|
ACKNOWLEDGNIENTS
Tjtis\ork\nsl)llrtiru\'sllPt)()|i|(ll)r'tli(,GltrntAgo1l(Ioftll|
(sr^rrt \o
l A . \ 1 0 { i 5 2 0 j t ) .l ) ! r h f L L s t i t u t i r ) r r l R { s c n | ( l i P l d r o f r h c L N r i i r l h , o f S ( i ( , r t i f i . l l n f r u r ! , i j
{IRp
N.. AVr) 210('51)ilr).{u(l l)! thf Elropfru Cinnuis$n,! tjtL fllllx$ork
],o.rhull r, \LS.t AD\jENTURE
A,Jiritv
tlrr)llgli th. Pro'{\i ATO]t 3D {50it1ri)2).
REFERENCES
1'A'-{sllknI''],]1DZn!1zi(.,I'E'Bj(x'kilolln.lllllS'c'llll..()|)s.|\'ntn'iofllsiliAl('lX'nnl1]|l(li(\|o|c.
( ) t ) t i ( n lt h p f i ) r ' , l i ( 1 . ( t r n
l)irrri{l.s. Olt. Ltt.71.1)t).2l{S !90. tfti(j.
i l i f ( r n o g r , r l ) l ^ . . 1 | 1 , l P h r \ . L r t t . 8 3 . p p . 5 i t 1 f i5 i t r . . 2 t ) 0 r t .
: i . X . S n n ! n l r r n d S . l L B l ( t r k . - B n n ( , 8 n a l a p p t i ( . n i n , r so f o t ) i i ( . r t t { 1 \ , 7 f r s . ) h r .
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P.oc.ol SP|EVo.5958 59582p-7
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