Twisted chains of resonant particles: Optical polarization control, waveguidance, and

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1 Twisted chains f resnant particles: Optical plarizatin cntrl, waveguidance, and radiatin D. Van Orden, Y. Fainman, and V. Lmakin *Department f Electrical and Cmputer Engineering, University f Califrnia, San Dieg, La Jlla, CA , dvanrden@physics.ucsd.edu, vitaliy@ece.ucsd.edu Abstract: Linear chains f metallic nanparticles that are sequentially rtated abut the chain axis display interesting plarizatin sensitive ptical prperties. Such twisted chains psses and extend prperties f chiral gratings and general peridic gratings. They are characterized by high anistrpy and plarizatin sensitivity, and have subwavelength transverse dimensins. These structures are shwn t supprt transverse mdes with distinct prpagatin wavenumbers and radiatin prperties, including slw (bund) and fast (radiative) mdes. They als have stpbands f different types, resulting frm cupling between distinct transverse mdes as well as cupling with different higher-rder diffractin mdes. Optical plarizatin cntrl is imprtant t the develpment f many integrated ptical systems. Miniaturizatin f mdern and future light-wave circuits demands develpment f subwavelength devices, such as radiatrs, filters, and cuplers that are sensitive t the plarizatin f light. Existing ptically active plarizatin devices include bulk media, such as liquid crystal media [1-2], surface structures [3], and helical waveguides and fiber gratings [4-5]. Amng thse structures that supprt prpagatin and waveguidance, liquid crystals have relatively weak anistrpy, while helical fibers and waveguides cannt easily cnfine light t small dimensins, thus cmplicating dense integratin. Realizing subwavelength ptically active waveguiding and radiating structures is imprtant t advance the develpment f dense nanphtnic systems. In this Letter we intrduce twisted particle chain structures with strng plarizatin sensitive prpagatin and radiatin prperties. The structures are cmprised f linear arrays f 1

2 subwavelength metallic particles that are rtated sequentially abut the chain axis (Fig. 1). Such structures have several unique prperties. Like all peridic nanparticle chains [6-1], twisted chains supprt traveling waves that are mediated by strng interparticle interactin facilitated by plasmnic resnances. Intrducing a sequential rtatin t each nanparticle, hwever, cuples the therwise independent transverse wave mdes, resulting in mdes with distinct plarizatin behavir. As a result, these structures (i) share certain plarizatin prperties with chiral gratings, (ii) psses diffractin prperties f general peridic gratings, (iii) have transverse dimensins f subwavelength size, and (iv) achieve high anistrpy and plarizatin sensitivity due t the particle resnances. These prperties can have imprtant applicatins t nan-scale ptical devices. The presented analysis is als imprtant t understand prperties f general waveguiding and radiating structures. We cnsider an infinite peridic chain f nnspherical resnant nanparticles that are sequentially rtated abut the chain axis by a cnstant angle (Fig. 1). The chain lies n the z axis and has spacing d. The particles have plarizability tensr α whse transverse principal values xx and yy are nt equal. Twisted arrays similar t thse in Fig. 1 can be fabricated fr example using a frce mediating plymer t bend silicn nanwires, which can be cated with metal [-]. In the arrangement f Fig. 1 the lngitudinal traveling wave mdes are independent and Deleted: is are nt affected by the sequential rtatin. We therefre restrict analysis t the transverse traveling wave mdes and cnsider nly the transverse field cmpnents. A cmplete analysis f the radiatin and prpagatin prperties f mdal fields supprted by the structure in Fig. 1 can be perfrmed using the discrete diple apprach with the dyadic peridic Green s functin f the twisted array. In this apprach, the chain elements are assumed t be small cmpared t the free-space wavelength. T find the surce-free mdal fields all Deleted: Althugh the structure as described wuld be very difficult t realize in an experimental setup, its general prperties are representative f any resnant array with twisted bundary cnditins. Similar structures culd include arrays f metalcated silicn pillars n a surface bent as in [], r even arrays f half wavelength diples in the micrwave regime. In bth cases surface interactins wuld have t be accunted fr. external surce fields are set t zer. The diple mment f the particle in the zerth unit cell f the chain is therefre given by p α E, where E is the electric field generated by all the rest 2

3 f the particles in the chain. The diple mment f all the chain elements have the same magnitude, and are related t p via rtatinal matrices, determined by the chain twist angle, and via a phase shift between the elements that is determined by the mdal wavenumber. The electric field E is fund as a superpsitin f cntributins frm the diple mments as E Grt p, where rt G is a rtated dyadic peridic Green s functin accunting fr the array s elements with rtatin. This apprach leads t the fllwing self-cnsistency relatin G rt ( I α G ) p, rt 1 Gxx Gxx i( Gxy Gxy ) Gxy Gxy i( Gxx Gxx ). 2 Gyx Gyx i( Gyy Gyy ) Gyy Gyy i( Gyx Gyx ) (1) Field Cde Changed Here, the rtated dyadic Green s functin is fund via the elements f the matrices G G ( k, d), where G is the dyadic peridic Green s functin f an infinite array f parallel diples with the bservatin pint taken t be the rigin, and the surce cntributin f the particle lcated there remved s as t avid self-interactin. It is given by ik z nd 2 ik r nd zˆ z n, n G( k, k ) (1 4 ) e ( k I) e rndz ˆ, where k 2 is the r Frmatted: Highlight Frmatted: Fnt: 11 pt, Lwered by 5 pt free-space wavenumber. This lattice sum is slwly cnvergent but can be rapidly cmputed using certain spectral techniques, e.g. as in [11]. The self-cnsistency equatin (1) can be slved fr the Deleted: (1) (generally cmplex valued) mdal wavenumber and the crrespnding rati p p. x y The resulting mdal fields have the same translatin and rtatin symmetry as the particle plarizatin stateserrr! Reference surce nt fund., thugh the symmetries are cntinuus, nt discrete. The mdal electric field may be expressed in tw alternative frms: zd cs( ) sin( zd) i z 1 i 1 R z il z E zz Ex Eye ER e EL e (2) sin( zd) cs( zd) i i Field Cde Changed where R d and L d. The tw expressins in Eq. Errr! Reference surce nt fund.(2) highlight different prperties f the traveling wave Deleted: (2) 3

4 mdes. The first expressin describes the mde prpagatin in terms f tw rtating cmpnents and gives the field in the rtated crdinate system. The secnd expressin, n the ther hand, gives the mdal field in terms f a left handed circularly plarized (LCP) and a right handed circularly (RCP) cmpnent in the Cartesian crdinate system, and represents the fields as bserved in a labratry reference frame. The crrespnding RCP and LCP wavenumbers are given by R and L, respectively. This latter expressin reveals the radiatin prperties f the Deleted: HC Deleted: HC chain. The self-cnsistency equatin (1) may be slved numerically t find fur surce-free Deleted: (1) slutins in the first Brilluin zne (with respect t the peridicity d ), representing tw independent mdes prpagating in each directin. The mdes with psitive grup velcity have Deleted: se wavenumbers and, fund near k d and k d, respectively, while thse with wavenumbers and have negative grup velcity. Because f the structure peridicity d, the mdal fields can be represented as a sum f an infinite number f diffractin (Flquet) mdes whse wavenumbers 2md (with m an integer) als slve Eq. (1) m Deleted: and Deleted:, Deleted: representing prpagatin in the ppsite directin Deleted: (1) and crrespnd t surce-free mdal fields. The mdal fields can exhibit several types f behavir, resulting frm the presence f the twist and the discrete peridicity, including stpbands f tw types, prpagatin withut radiatin lss, and radiatin f tw types. Stp-bands frm fr certain chain gemetries and frequency ranges when tw distinct mdes with cunter prpagating energy cuple strngly. One way this Deleted: ppsite grup velcities can ccur is thrugh cupling f ne f the fur traveling wave mdes mentined abve t a higher diffractin rder f anther ne. Fr example, a stpband may frm thrugh cupling f first rder diffractin mdes when Re{ } 2 d Re{ } r Re{ } 2 d Re{ }. The strng mde cupling results frm the interplay between the effects f the twisting with angle and the peridicity d. Such interplay is absent in rdinary chiral gratings r peridic 4

5 (nn-chiral) gratings. Anther way a stpband may frm is thrugh direct cupling between the and mdes that ccurs fr sufficiently large d when Re{ }. Figure 2(a) shws the dispersin curves fr a chain f gld prlate spherids embedded in a silicn dixide medium with spacing d 72nm and twist angle 18. The curves are fund by tracking the real parts f and ver a range f values f the free-space wavenumber k fr which waveguidance is supprted. Tw sets f curves are shwn: ne fr realistic lssy gld particles whse permittivity is given by the Drude mdel with parameters chsen t match thse given by Palik [12], and the ther fr lssless Drude gld particles with a vanishing damping cnstant. In bth cases the mde cuples t the first rder diffractin mde f, fund at 2 d. In the lssless case, fr which is a purely real number, it frms a clear stpband, lcated at d, where the tw slutins merge. In the lssy case the tw slutins have imaginary parts with ppsite signs and remain separated; n sharp band edge frms, Deleted: althugh the cupling is evident, althugh the cupling is evident frm the way the dispersin curve fr bends back away frm the lssless stpband pint, indicating a change in grup velcity,. Figure 2(b) shws the dispersin diagram fr lssy and lssless chains identical t thse cnsidered in Fig. 2(a) except fr a larger twist angle, which is nw 85. Fr the lssless chain a stp-band frms right at the rigin, while fr the lssy chain the curves bend away frm each ther. The prpagatin and radiatin prperties f twisted chains are best described by the secnd expressin in Eq. (2)Errr! Reference surce nt fund.. First cnsider the case f Deleted: The presence f a stp-band culd be explited t create waveguiding chains with a tunable phase velcity r resnatrs. Deleted: (2) sufficiently small d fr which higher rder diffractin mdes are nt in the prpagatin range. Because d k, the mde is a slw wave that des nt radiate and whse fields decay expnentially transverse t the chain. This mde can is best excited by a surce with an evanescent spectrum. The mde, in cntrast, is cmprised f an RCP wave with wavenumber and an LCP wave with wavenumber L R d k d that can be either in 5

6 the evanescent r prpagating range. Fr small t mderate twist rates, L k and the mde is a leaky (fast) wave whse LCP cmpnent radiates ut f chain at an angle rad with the chain ( z ) axis. The radiatin angle is apprximately determined by the phase matching cnditin k cs( ) Re{ } d. It can be shwn that the radiated beam has left handed elliptical rad plarizatin with ellipticity (i.e. majr t minr axis rati) given by e sec( rad ). It fllws frm reciprcity that this mde may be excited by a LCP far-field surce r by a surce having a RCP evanescent spectrum (e.g. via a prism). If the particle spacing d is large enugh fr diffractin mdes t enter the radiatin range, case, the mde may als be leaky. As a result, the chain will radiate int multiple beams at distinct angles. Fr example, assuming that nly ne diffractin rder participates in the radiatin prcess, the chain will emit tw additinal radiatin beams at angles kcs( ) Re{ 1} d. rad The predicted radiatin characteristics f traveling wave mdes may be verified by Deleted: The separatin between the radiatin angles is determined by the twist angle; e.g. this separatin can be very small, frming a beam dublet, r large. simulating a finite twisted chain. A finite structure cannt be analyzed using the peridic Green s functin, but the Green s functin fr a single diple surce may be used t find the interactin amng all the chain elements and slve fr the induced the diple mment f each nanparticle in respnse t an external surce. Cnsider a chain f 2 particles with the same parameters as in the infinite case in Fig 2. The chain is excited by an RCP lcalized surce placed n the z axis a distance d 72nm frm the chain. Figure 3(a) shws the radiatin pattern as a functin f the bservatin angle with respect t the z axis far frm the chain, where the far-field dminates. Fr this dense array, the radiatin results slely frm the chain twist. The pattern shws a clear peak at the angle rad at which radiatin frm the traveling wave mde is predicted fr an infinite structure. The chain acts as a nanscale leaky-wave antenna whse radiatin angle and plarizatin may be tuned by varying the twist rate d. Figure 3(b) shws the radiatin pattern fr a chain with twist angle and a larger peridicity d 181nm. Fr this large spacing 48 6

7 Flquet mdes lead t radiatin fr the therwise slw mde. Here, tw peaks are bserved with radiatin angles 1 and 2 agreeing with the cnditins kcs( 1) Re{ } d 2 d and 2 k cs( ) Re{ } d 2 d. In cnclusin, we have analyzed the traveling wave mdes f a linear chain f sequentially rtated nanparticles. We have shwn that tw mdes are fund prpagating in each directin and that fr sufficiently small spacings, ne f these mdes is a slw wave bund t the chain while the ther is a fast wave that radiates. Furthermre, the structure s dispersin relatins exhibit stpbands f tw kinds: thse assciated with cupled diffractin mdes and thse resulting frm ppsitely prpagating zer-rder mdes. These structures may have applicatins as subwavelength plarizatin sensitive ptical filters and antennas. This wrk is supprted by DARPA NACHOS prgram, and NSF CIAN center. 7

8 1. H. L. Ong, "Wave prpagatin in chlesteric and chiral smectic-c liquid crystals: Exact and generalized gemetrical-ptics-apprximatin slutins," Phys. Rev. A 37, (1987). 2. C. Oldan, E. Miraldi, and P. Taverna Valabrega, "Dispersin relatin fr prpagatin f light in chlesteric liquid crystals," Phys. Rev. A 27, (1983). 3. A. Papakstas, A. Ptts, D. M. Bagnall, S. L. Prsvirnin, H. J. Cles, and N. I. Zheludev, "Optical Manifestatins f Planar Chirality," Phys. Rev. Lett. 9 (23). 4. V. I. Kpp, V. M. Churikv, J. Singer, N. Cha, D. Neugrschl, and A. Z. Genack, "Chiral Fiber Gratings," Science 35, (24). 5. G. Shvets, "Optical plarizer/islater based n a rectangular waveguide with helical grves," Applied Physics Letters 89, (26). 6. M. Quinten, A. Leitner, R. Krenn, and F. R. Aussenegg, "Electrmagnetic energy transprt via linear chains f silver nanparticles," Opt. Lett. 23, (1998). 7. M. L. Brngersma, J. W. Hartman, and H. A. Atwater, "Electrmagnetic energy transfer and switching in nanparticle chain arrays belw the diffractin limit," Phys. Rev. B 62, R (2). 8. W. H. Weber and G. W. Frd, "Prpagatin f ptical excitatins by diplar interactins in metal nanparticle chains," Phys. Rev. B (Cndensed Matter and Materials Physics) 7, (24). 9. A. Alù and N. Engheta, "Thery f linear chains f metamaterial/plasmnic particles as subdiffractin ptical nantransmissin lines," Phys. Rev. B 74, (26). 1. D. A. Van Orden, Y. Fainman, and V. Lmakin, "Optical waves n nanparticle chains cupled with surfaces," Opt. Lett. 34, (29). 11. D. Van Orden and V. Lmakin, "Rapidly cnvergent representatins fr 2D and 3D Green s functins fr a linear peridic array f diple surces," IEEE Trans. Ant. Prpagat. 57, (29). 12. E. D. Palik, Handbk f Optical Cnstants f Slids (Academic Press, 1998). 8

9 Figure 1. (Clr nline) A twisted chain f nanparticles with spacing d and twist angle. Figure 2. (Clr nline) (a) Dispersin relatins f the and mdes fr chains f lssless and lssy gld ellipsids with spacing d 72nm, twist angle 18, and majr and minr axes f lengths 34nm and 17nm, respectively. (b) Dispersin relatins f the mde fr the same chain, but with twist angle 85. Figure 3. (Clr nline) (a) Far-field radiatin intensity bserved a distance 96 m frm a chain f 2 gld ellipsids with spacing d 72nm, twist angle 18, excited by a lcalized RCP surce with wavelength 527nm (b) Radiatin pattern frm the same chain but with spacing d 181nm, twist angle 48, excited by an LCP lcalized surce. 9

10 Figure 1 y x z Figure 2 k d lssy lssless ( a ) ( b) d d Figure 3 nrmalized electric field intensity ( a ) ( b) angle (rad) Radiatin Angle angle (rad) 1