Optics Communications

Optics Communications 84 () 43 436 Contnts lists availabl at ScincDirct Optics Communications journal hompag: www.lsvir.com/locat/optcom Scattring forcs in th focal volum of high numrical aprtur microscop objctivs gnacio glsias a,, Juan José Sánz b,c a Dpartamnto d Física, Univrsidad d Murcia, Campus d Espinardo (COyN Bldg.), E-3, Murcia, Spain b Dpartamno d Física d la Matria Condnsada, Univrsidad Autónoma d Madrid, E-849 Madrid, Spain c Donostia ntrnational Physics Cntr (DPC), Paso Manul Lardizabal 4, E-8 Donostia-San Sbastian, Spain articl info abstract Articl history: Rcivd 4 Novmbr Rcivd in rvisd form January Accptd 3 January Availabl onlin Fbruary Radiation prssur is not th only sourc of scattring forc in th focal rgion of a microscop objctiv. Dpnding on th numrical aprtur and th polarization charactristics of th light at th ntranc pupil, th forc mrging from th spin dnsity may rprsnt a fundamntal contribution to th total forc xprincd by small (Rayligh) particls whn using high rsolution objctivs. Th prdictd and xprimntally obsrvd strong asymmtry of th trapping potntial for linar polarization is shown to b rlatd to non-consrvativ spin curl forcs. W dmonstrat th charactristics of this forc fild whn using linar polarizd light with circular and annular pupils and in th radial and azimuthal polarization structurs. Elsvir B.V. All rights rsrvd.. ntroduction Light can b usd to manipulat small micron and nano-sizd particls in soft mattr, allowing physics and biology xprimnts to b dsignd to obtain information about th particls or th surrounding nvironmnt. n ths xprimnts, prcis knowldg of th optical forcs is ssntial. Usually, th forcs ar gnratd by light bams focusd by microscop objctivs [,], which ar convnint optical systms for trapping or inducing movmnts of particls insid sampls from a distanc. Thr ar two typs of forc gnratd by light on th focal rgion of a microscop objctiv: th trapping forc and th scattring forc. Th first mrgs from th gradint of th optical intnsity distribution and is rsponsibl for th attractiv potntial xprincd by dilctric particls [3]. n th litratur, radiation prssur (proportional to th Poynting vctor), is usually [4,5] statd as th origin of th scattring forc. Dspit this, in high NA objctivs th scattring forc has also bn approximatd, apparing to b dpndnt not on th radiation prssur but on th intnsity of th optical fild [4,6]. Howvr, approximation is not ndd whn considring small Rayligh particls in an optical fild. n this cas, th scattring forc can b calculatd xactly [7,8] and th radiation prssur mrgs as th sourc of th scattring forc, togthr with an additional trm which, dpnding on th particular optical systm, may b crucial: th Corrsponding author. E-mail addrss: iic@um.s (. glsias). forc gnratd by th curl of th spin angular momntum of th light fild. This scond contribution has bn studid for optical filds in optical lattics [8 ] and, in gnral, du to th curl natur of th spin trm, it is most rlvant for th study of particl dynamics in nonconsrvativ forc filds [4, 3]. n this work, w show that this spin trm also plays a significant rol in th total scattring forc in th focal volum of high numrical aprtur (NA) objctivs. Spcifically, on a small dilctric sub-wavlngth particl, th total forc gradint forc plus scattring forc can b writtn as, n R E x E x + E y E y + E z E o z whr E is th lctrical fild and is th polarizability, which is rlatd with th inducd lctric dipol by p = E. For a divrgnc-fr fild, th scattring componnt can b split into two trms radiation prssur plus spin dnsity forc as [8]: F + F sd = σ c hsi + c ð h L siþ whr σ = kf gis th total (xtinction) cross sction (s Rf. [4] for an xtnsion to magntodilctric particls) and k=π/ is th fr spac wavnumbr. n th abov quation, L s is th tim avragd spin dnsity and S is th tim avragd Poynting vctor or radiation prssur, i.., hl s i = F ε n o E E 4ωi ðþ ðþ ð3þ 3-48/$ s front mattr Elsvir B.V. All rights rsrvd. doi:.6/j.optcom...9

. glsias, J.J. Sánz / Optics Communications 84 () 43 436 43 whr ε is th lctric prmittivity of fr spac and ω th optical angular frquncy, and n o hsi = R E H whr H is th magntic fild. Th positiv or ngativ sign in Eq. (3) dpnds on th chosn tim dpndnc xp{ iωt}, which also mans that th corrsponding sign should b in th Maxwll quation E=±iωμ H. n addition to clarifying th physical origin of th scattring forc, th us of Eq. () rathr than Eq. () (subtracting th part corrsponding to th gradint forc), has th additional advantag of involving ral functions, which simplifis numrical computations whn daling with complicatd fild distributions. Whn th light is linarly polarizd, th spin dnsity of Eq. (3) is constant. Thrfor, th curl in Eq. () is idntically zro and thr is no spin trm F sd. Howvr, whn considring a microscop objctiv, vn using a linarly polarizd incoming light bam, a vctor rotation is inducd by th lns and additional fild componnts ar gnratd. For low NA, th light aftr th lns can still b considrd as linarly polarizd and th curl trm ignord, laving radiation prssur as th only sourc of th scattring forc. With high NA objctivs, this rotation of th lctric (and magntic) vctor is significant for numrous propagation dirctions. t follows that th spin dnsity of th light aftr th microscop objctiv cannot b considrd constant and th curl scattring forc trm cannot b discardd. Consquntly, to find th dtaild charactristics of this additional forc trm, it must b studid and compard with th radiation prssur in ordr to obtain a complt pictur of th scattring forcs at play in focal volums. Th rst of this papr is dvotd to discussing this in particular cass.. Scattring forcs in th focal volum.. Numrical computation of th forcs using th Dby diffraction intgral Th first stp is to obtain th complt lctric and magntic fild vctors in th focal volum. For this puos, lt us considr th gomtry sktchd in Fig.. For an incoming light bam, E i(kz ωt) x of uniform amplitud and linarly polarizd along th x-axis, following Fig.. Dfinition of coordinats and angls. E inc indicats th polarization dirction of th incoming lctrical fild; E is th lctrical fild aftr th rotation inducd by th lns; x, y and z ar th coordinat axs cntrd at th focal point; θ is th angl with th optical axis of a givn propagation vctor; ψ is th angl for th polar coordinats in th focal plan. Th angl dfins th numrical aprtur (NA=sin ). ð4þ Lommls approach [5], th fild, onc it has bn propagatd to th microscop focal rgion r =(x, y, z)=(ρ cos ψ, ρ sin ψ, z), can b xprssd in trms of th intgrals, and as [6] Er ðþ= ikf ikf E ð + cos ψ; sinψ; i cosψþ Hr ðþ= ikf E ikf ð sin ψ; cosψ; i sinψþ η whr, for simplicity, it has bn p assumd an indx of rfraction, f is th focal lngth, η = μ c = μ = ε is th vacuum impdanc and ðþ= r ðþ= r ðþ= r p cosθ p cosθ p cosθ sinθð+cosθ ÞJ ðkρ sinθþ ikz cosθ dθ sin θj ðkρ sinθþ ikz cosθ dθ sinθð cosθ ÞJ ðkρ sinθþ ikz cosθ dθ: Thus, ths intgrals hav to b numrically valuatd for ach fild point. As an altrnativ approach, th Dby diffraction intgral can b considrd as a Fourir transform [7,8], xprssing th fild componnts aftr propagation in th focal rgion as: Er ðþ= ẼðkÞ ikr dk xdk y dk z ; Hr ðþ= Hðk Þ ikr dk x dk y dk z ð7þ ðπþ 3 ðπþ 3 whr Ẽk ð Þ and HðkÞ ar th amplitud of th lctric and magntic fild componnts rspctivly bfor th propagation dfind in th rciprocal k-spac as functions of th wav vctor k = k x ; k y ; k z.n Eq. (7), Ẽk ð Þ = ikf ikf ðπ=kþ δðjkj kþe ; HðkÞ = ikf ikf ðπ=kþ δðjkj kþh ð8þ ar non-zro for a sphrical cap on th Ewald sphr with radius k and a surfac ara dtrmind by th maximum angl of convrgnc to th focus,, which srvs to dfin th microscop NA as sin, assuming a rfraction indx of on. Eq. (7) can b usd to comput th fild componnts E i ðþand r H i ðþnumrically, r using a tridimnsional fast Fourir transform aftr th discrtization of Ẽ i ðkþ and H i ðkþ. W us Eq. () to obtain th scattring forc filds, F sd and F, using th abov mthod to comput th lctric and magntic filds. W considr th simpl cas of an incoming bam of linarly polarizd light along th dirction dfind as th x-axis with =5 nm for a microscop objctiv charactrizd by =7. Ẽ i ðkþ and H i ðkþ ar sampld to fit matrics with 3 lmnts, padding th sphrical caps by mbdding th sampld valus in null matrics of 4 3 lmnts that ar transformd using a thr-dimnsional fast Fourir algorithm. Fig. shows bidimnsional contour plots corrsponding to svral cuts of a 5 3 voxl rgion of th scattring forc fild matrics cntrd at th focus (r =(,, )). Fig. 3 shows schmatically som charactristic forc vctors on th xz and yz-plans to hlp in th intrtation of th numrical rsults. t can b obsrvd that th axial componnts F z and F sd z (in panl A of Fig. ) ar th major contributors to th total scattring forc compard with th latral componnts F x and F sd x (in panl B of Fig. ). Th radiation prssur axial componnt, F z, in th top row of Fig. in panl A, shows th rotation symmtry sd around th optical axis (z-axis), which is absnt from th curl forc F z shown in th bottom row. This componnt also has th pculiarity of pointing backwards in th opposit dirction to that of th light propagation at points around th optical axis, raching a maximum valu on th optical axis. Th transvrsal x-componnts of F and F sd ð5þ ð6þ

43. glsias, J.J. Sánz / Optics Communications 84 () 43 436 Fig.. Panl A, sctions of th longitudinal componnts of th scattring forc filds in th focal volum of a microscop objctiv ( = 7 ) illuminatd with an x-polarizd bam ( = 5 nm). Panl B, th corrsponding transvrsal componnts p(in th polarization dirction); th dottd lins indicat approximat axs of symmtry. Th plots rprsnt aras of.6.6. Th amplitud of th lctrical fild is je j = = f σ. Panls A and B hav th sam units, th color scal is adaptd to th rspctiv rangs. (in panl B of Fig. ) ar charactrizd by th quivalu llipsoidal distributions around th optical axis, with comparabl amplituds but opposit signs. Consquntly, th x-componnt of th total scattring forc is vry small. n Fig. (), F y is not shown, sinc it has an amplitud distribution quivalnt to F x onc rotatd 9 (in th sns of incrasing ψ) around th optical axis. W also choos not to show th F ysd componnt bcaus of its small valu compard with th rst; th y-componnt mrging from th radiation prssur is thn th only transvrsal componnt of th total scattring fild. Fig. 3 schmatically summarizs th prvious rsults, dpicting charactristic vctors and tangnt curvs. To facilitat th visualization, th color of th vctors corrsponds to th color bars in Fig.. Panl (A) shows vctors corrsponding to th symmtrical radiation prssur fild on th xz and yz-plans: th transvrsal componnts bcom rlvant only at a crtain distanc (indicatd as a dottd circl) from th focal point, dviating th forc vctors with rspct to th optical axis dirction. Panl (B) shows th curl forc whr, givn th small valu of th curl forc on th yz-plan, no vctors hav bn dpictd apart from th on along th optical axis. Panl (C) shows th total scattring forc, th lins (tangnt to th forc vctors) corrspond to trajctoris (approximatly straight lins on th polarization plan and curvs on th pndicular plan) that will follow scattrd mtallic particls (not subjct to th gradint forc) in an ovrdampd rgim. with F = ( ) πf σ ε je j which is symmtric around th optical axis. Also, on th plans dfind by ψ = (xz-plan) and ψ = 9 (yz-plan), th radiation prssur componnts F y and F x ar absnt. Ths analytical rsults agr with th numrical rsults shown in Fig.. At th focus, th (non-consrvativ) spin dnsity forc is also normal to th focal plan. Aftr som algbra, Fzsd = F cos ψℜ Lt us analyz th scattring forcs in th focal plan z =. Th radiation prssur forc (normal to th focal plan) can b obtaind dirctly from th Poynting vctor and is simply givn by )!( " # # ð + Þ i ðkρþ ðkzþ z = ðþ!( " ) # F sin ψℜ ð Þ kρ z = which ar xact rsults. To gain insight into th numrical rsults, th following approximat xprssions can b usd for th rlvant intgrals. At th focal plan (z =, r = ρ x + y ), can b approximatd as ðρ; z = Þ C.. Th forcs xprssd in trms of Bssl intgrals ðþ J ðkρ sinþ kρ sin ðþ whr C = ð; z = Þ ð3þ and Fz = F j j j j z= ð9þ ð; z = Þ = ( p ) 6 p5 cos3 + cos 5 3 5 ð4þ

. glsias, J.J. Sánz / Optics Communications 84 () 43 436 433 Fig. 3. Schmatic rprsntation of som charactristic forc vctors around th focus on th xz and yz-plans for an x-axis linarly polarizd incoming bam. (a) Th radiation prssur; (b) th spin dnsity forc; and (c) th total scattring forc. Th optical axis is th z-axis. Th continuous thin lins rprsnt curvs tangnt to th forcs. which is valid for high NA. For small NA, () sin and Eq. () coincids with th approximat xprssion for that appars in Chon t al. [9]. Eq. () is th Airy function of th paraxial approximation whr sin. Following th sam approach, w find a simpl xtnsion to th othr Bssl intgrals that appar in Eq. () J ðρ; z =Þ C ðkρ sinþ kρ sin ð5þ whr C is a constant chosn to match th maximum valu of J (kρ sin )/kρ sin to th xact maximum valu of (ρ;z=). ð6þ ðkρþ sinc 3C kρ sin ; ðkzþ i : ð7þ With this simpl ansatz, th radiation prssur trm is proportional to th fild intnsity in th focal plan and follows th typical Airy profil around th optical axis, C z F J ðkρ sinþ kρ sin F whras th curl forc is approximatd as F sd z F C C sin J C C ðkρ sinþ kρ sin 3 J ð kρ sin Þ kρ sin C J ðkρ sinþ! kρ sin cos ψ F C C sin J ðkρ sinþ kρ sin J ðkρ sin ðkρ sinþ J ðkρ sinþ ðkρ sinþ! Þ! ð8þ sin ψ: ð9þ Plotting this quation (blu and grn dottd lin in Fig. 4(a) corrctly rproducs th distribution of th curl forc in Fig. on th focal plan (xy-plan) along th x-axis and y-axis. (a) (b) 5 8 5..4.6.8. 6 4-5...4.6.8. Fig. 4. (a) Th radiation prssur on th focal plan for a lns with =8, and th spin dnsity forc at ψ=9 and at ψ=. (b) Th total scattring forc for =8 and =5 at ψ= and th normalizd intnsity.

434. glsias, J.J. Sánz / Optics Communications 84 () 43 436 polarization dirction. As Fig. 4(b) shows, this distribution is approximatly rplicatd by th normalizd intnsity which bcoms a good approximation of th total scattring. As Fig. 4(b) also shows, th fit improvs whn th NA is rducd. For linar polarization along th x-axis, th rsults can b summarizd as: 5 s Fx ðrþ Fys ðrþ Fy ðrþ..4.6.8. ðþ s Fz ðrþ ðrþ -5 Fig. 5. Th radiation prssur on th focal plan at ψ =, curl forc, total scattring and normalizd intnsity along th axial dirction for radial polarization (doughnut mod) and an objctiv with = 7. whr Ex + Ey + Ez and Fzs Fys. This dpndnc on th intnsity of th main componnt of th scattring forc concurs with th xprssion for th total scattring forc givn by Rohrbach [6] for a linar polarizd bam using th first-ordr Bohr approximation. 3. Scattring forcs in radial and azimuthally structurd pupil polarization Using th filds in Eq. () to comput th forcs is tdious, givn th nd to numrically calculat th dfinit intgrals with complx krnls that finally appar in th xprssions. Dspit this, diffrnt axs and dirctions wr calculatd, and that which follows is a dscription of svral particular cass. n Fig. 4(a) th symmtric radiation prssur (rd dottd lin) and th asymmtric curl forc (blu and grn lins) has bn rprsntd for a vry high NA, rplicating th bhavior of th rsults obtaind with th Fourir transform mthod for both axs. t bcoms apparnt that th curl forc rprsnts a significant contribution to th total scattring forc. Th rlativ amplitud of th two contributions dpnds on th NA. Th main ffct of th curl forc is to rduc th forward scattring forc along th optical axis and to broadn th forward scattring around th optical axis from zro at ψ = 9 to its maximum in th xy Radial and azimuthally polarizd bams ar important xampls of non-uniform polarization in microscop objctivs [6,,]. Ths structurs can b gnratd by diffrnt optical systms,.g., th suposition of two linar polarizd highr ordr lasr mods. n this cas, for radial polarization and in trms of th Bssl functions, th lctric and magntic filds around th objctiv focus ar givn by Rf. [6] E= H= ikf ikf E i Þ cos ψ; ðð i Þ sinψ; 4 ikf ikf E $ i ð + 3 Þ sinψ; ið + 3 Þ cosψ; Þ η ðþ xz xy xz Fig. 6. Normalizd intnsity distribution, E, for pur radial (uppr row) and pur azimuthal (bottom row) bam polarizations and schmatic rprsntation of th lctrical fild vctor orintation at th pupil plan for a crtain radius. Th rst of th paramtrs ar th sam as in Fig..

. glsias, J.J. Sánz / Optics Communications 84 () 43 436 whr p 3 ikz cosθ dθ cosθ sin θj ðkρ sinθþ p ikz cosθ ðr Þ = dθ cosθ sin θð + 3 cosθþj ðkρ sinθþ p ikz cosθ ðr Þ = dθ: cosθ sin θð cosθþj ðkρ sinθþ ðr Þ = ðþ n this cas, th axial componnt (th main componnt of th radiation prssur) at th focal plan bcoms * * * * Fz = F *j jj 3*j jj + ð3þ z= whras th sam componnt of th curl forc is sd Fz = 4F k k!! + ρ ðkρþ : z= ð4þ Fig. 5 plots Eq. (3) (solid blu lin) and Eq. (4) (solid rd lin) for a microscop objctiv with = 7. Th radial polarization compltly altrs th scattring forcs. Th radiation prssur acquirs an annular distribution, whras th curl forc incrass in magnitud and is dirctd, contrary to th linar polarization, in th forward dirction, bcoming th only forc on th optical axis. Th azimuthal polarization can b obtaind in th sam way gnrating, at th focal volum, th filds givn by Rf. [6] ikf ikf E i + 3 cosψ; i + 3 sinψ; E= ikf ikf E i sinψ; i cosψ; 4 H= η As bfor, a numrical computation using th Fourir transform mthod can b carrid out using, to simplify th implmntation, pur radial and azimuthal polarization structurs on a circular aprtur rathr than th doughnut mod combination. Accordingly, th lctrical fild amplitud at a clar circular pupil is chosn uniform, with a polarization stat varying for ach point in th Jons matrix formalism []: J ðϕþ = + cosðϕ + ϕ Þ sinðϕ + ϕ Þ, from which can b dirctly dducd that th spin curl forc bcoms null, whras th radiation prssur dos not chang compard with th radial polarizd bam. ð6þ whr, ϕ = for azimuthal and ϕ = π/ for radial polarization, as schmatizd on th right of Fig. 6, with th corrsponding focal volum intnsity distribution apparing in th lft panl. This simplifid structur rproducs th main charactristics of th lctric and magntic fild distributions but not th compactnss [] gnratd by annular-lik distributions. Fig. 7 shows th computation of th radiation prssur and th curl forc for th radial polarization. Sinc th intnsity and forc fild componnts hav rotational symmtry around th optical axis for th radial (and azimuthal) polarization, w do not show th yz-plans in Figs. 6 and 7. n Fig. 7 it can b obsrvd that th radiation prssur has an annular structur and that th axial componnt of th curl forc is gratr than th corrsponding componnt of th radiation prssur. Ths rsults for pur radial polarization agr with thos for th annular radial polarization of Fig. 6. So, for th cas of radial polarization, it can b statd that, Fxs ðrþ = Fys ðrþ = Fx ðrþ + Fxsd ðrþ ð5þ 435 Fzs ðrþ ðrþ: ð7þ Th numrical computation for th azimuthal polarization is not shown: th radiation prssur is qual to th annular distribution gnratd by th radial polarization and th curl forc is ngligibl (dspit th fact that th valus obtaind by th numrical Fig. 7. Panl A, sctions of th longitudinal componnts of th scattring forc filds for a pur radial polarizd bam for = 7. Panl B, th transvrsal x-componnt of th scattring forcs. Othr paramtrs ar th sam as in Fig..

436. glsias, J.J. Sánz / Optics Communications 84 () 43 436 4 3 - - -3..4.6.8. Fig. 8. Th radiation prssur on th focal plan at ψ =, curl forc, total scattring and normalizd intnsity along th axial dirction for linar polarization for an objctiv with an annular pupil with =6 =7. computation ar not strictly zro, th amplitud is not nough to caus a contour lvl jump whn adapting th scal to th radiation prssur data). So, th scattring forc fild now complis with, 5. Conclusion W hav shown that th forc that ariss from th curl of th spin dnsity of th light xrtd on th sub-wavlngth particls in th focal rgion of a high NA microscop objctiv is a crucial contributor to th total scattring forc. For linar polarization, th symmtry of th radiation prssur around th optical axis is brokn by th curl forc, which xplains th asymmtry of th total scattring forc that has bn obsrvd [6,]. For th radial and azimuthal polarization structurs, th curl forc is fundamntal for modling th total scattring forc, vn with modrat NA: in th first cas, mainly to xplain th scattring forc on and around th optical axis; in th scond, to modl th annular forc distribution that appars to b gnratd only by th radiation prssur sinc th curl forc is absnt in this cas. W hav shown that in linar, radial and azimuthal polarizations, th intnsity distribution is a good approximation of th total scattring forc (F s ẑ). Howvr, w also show that this is not th cas with th linar polarization and annular pupils, whr it is prfrabl to considr th sum of th radiation prssur and th curl forc to modl th total forc. n conclusion, it has bn shown that th dcomposition of th scattring forc in th focus of high NA objctivs in trms of radiation prssur and curl forc, whil providing a physical insight into its origin, is a simplmodling mthodfor agnral pupil plan polarization structur. F s x ðþ= r Fs y r F s zðþ r ðþ r ðþ= F x ðþ r Finally, in Eqs. (7) and (8), as in Eq. (), F z s F y s. ð8þ Acknowldgmnts This work was supportd by th Spanish MEC through th Consolidr NanoLight CDS7-46 and Fundación Snca (Rgión d Murcia, Spain) grant 454/GERM/6MEC-CONSOLDER. 4. Linar polarization and annular pupil From th prvious rsults, it might b xpctd that th normalizd intnsity would b a good approximation for th scattring forc in th focal volum for a gnral polarization in th ntranc pupil plan of a high NA objctiv. Howvr this is not so in all cass, or at last th fit is not as good as might b xpctd, as can b sn in th xampl that follows. With linar polarization, vn with high NA microscop objctivs, clos-to-paraxial angular componnts contribut vry littl to th curl forc sinc th corrsponding spin dnsity in th focal volum is almost constant. Taking this into account, it is asy to s that th balanc btwn th radiation prssur and th curl forc is altrd with rspct to th uniform linar polarizd cas using annular pupils, which block paraxial light by mans of a cntral obscuration mask. Th objctivs with ths charactristics ar important for diffrnt applications sinc thy provid axially longatd intnsity distributions and qualizd lctric fild componnts which can b usd as xcitation probs [3]. Fig. 8 shows th rsult of th scattring forcs for an objctiv with an annular pupil illuminatd with an x-axis linar polarizd bam. Th graphs wr computd using Eq. (), and limiting th numrical intgration btwn angls =6 and =7. t can b obsrvd that th two forcs bcom mor qualizd in absolut valus on th optical axis. Howvr, contrary to th othr cass, th scald intnsity significantly ovrstimatd th scattring forc on th axis and undrstimatd it for points outsid th axis. Rfrncs [] D.G. Grir, Natur 44 (3) 8. [] A. Askin, J.M. Dzidzik, Scinc 35 (987) 57. [3] A. 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