Efficient excitation of plasmonic nanoantennas by tightly focused vector beams
Xiaorun Zang, Godofredo Bautista, Léo Turquet, Tero Setälä, Martti Kauranen, Jari Turunen
EEfficient excitation of plasmonicnanoantennas by tightly focused vector beams
Xiaorun Zang, ∗ , † , ‡ Godofredo Bautista, ‡ Léo Turquet, ‡ Tero Setälä, † MarttiKauranen, ‡ and Jari Turunen † † Institute of Photonics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu,Finland ‡ Photonics Laboratory, Physics Unit, Tampere University, P.O. Box 692, FI-33014,Tampere, Finland
E-mail: xiaorun.zang@uef.fi
Abstract
Hybrid modes of plasmonic nanoantennas are attractive to many applications, butexcitation of such modes by a conventional beam is not always efficient, and is some-times impossible, due to a field-profile mismatch between the incident light and thehybrid mode. We present a new approach for efficient hybrid-mode excitation by con-structing the incident light field in the basis of cylindrically polarized vector beams ofvarious higher-order spiral phases. Such basis vector beams are conveniently describedin the higher-order polarization states and Stokes parameters, with both concepts de-fined locally in polar coordinates, and visualized correspondingly on the higher-orderPoincaré spheres. Besides the spatially inhomogeneous polarization distribution, thehelical phase variation in a higher-order vector beam is also crucial for effective cou-pling with a hybrid mode. The constructed beam is tightly focused to concentratelight on sub-wavelength plasmonic nanoparticles. However, tight focusing causes po-larization transfers for each higher-order VB. Thus it is essential to further analyze the a r X i v : . [ phy s i c s . op ti c s ] S e p ocused field around nanoantennas so that an effective mode-matching field profile ismaintained. Through the boundary element method simulations, we demonstrate thatradially polarized beams of various orders are necessary to match and excite hybridmodes of distinct spectral resonances in our example of a radial tetramer, and highexcitation efficiencies are obtained when the incident beams are further tightly focused.The developed vector beam decomposition and the approach to analyze higher-orderpolarization states provide a general framework to shape the focal field for efficienthybrid-mode excitation in nanoantennas. Keywords hybrid mode, plasmonic nanoantennas, vector beam, focusing, polarization, higher-orderPoincaré sphere
Introduction
Hybridized plasmon modes, arising from optical near-field coupling in plasmonic nanoparticleassemblies are attractive for a variety of applications because of their unique spectraland radiation characteristics . Effective hybrid-mode excitation is crucial in achievingfunctionalities that rely on characteristics of such modes, which can be challenging as itsometimes requires a specially engineered incident light field when the conventional onesdo not work. For instance, the bonding and antibonding modes are formed in nanoroddimers , and the former can be efficiently excited by a plane wave at normal incidence,whereas the latter’s excitation is symmetry-forbidden for the same illumination but allowedby an obliquely incident plane wave , a localized emitter , or a tightly focused pulsed laserbeam . Another example is the use of cylindrical vector beams for the excitation of darkmodes in nanorod trimers or in other plasmonic clusters . In general, to efficiently excite ahybrid mode, it is essential to shape the field profile of the incident light so that it effectively2atches the hybrid mode’s profile, which can have a significant space-varying polarizationdistribution over the sub-wavelength range around the nanoparticles.Vector beams (VBs) with tuned spatial distributions of the polarization state addnew ingredients into the light-matter interaction. Recently, they have found a considerableamount of applications in diverse research fields, including optical microscopy , opticalcommunication , trapping , and nonlinear nano-optics . Typical examples arethe radially and azimuthally polarized VBs, where the electric field vectors point into theradial and azimuthal directions in the beam cross section, respectively . They are usuallyregarded as two fundamental zeroth-order instances of more general higher-order cylindricalVBs that can be geometrically illustrated on the higher-order Poincaré (HOP) spheres .Compared with the standard Poincaré sphere, every point on the HOP sphere correspondsto a higher-order VB which can possess an extra helical phase modulation in addition to thespatially inhomogeneous polarization distribution. General cylindrical VBs with arbitraryhigher-order polarization states have been recently extensively investigated, as they havebecome experimentally accessible .Here, we investigate the shaping of the incident light with an effective mode-matching fieldprofile through a decomposition into higher-order, cylindrically polarized VBs. We adopt theconcepts of higher-order polarization states, Stokes parameters, and HOP spheres in polarcoordinates on the basis of higher-order radial and azimuthal polarization states. Regardingthis adoption, each individual higher-order basis VB has a locally identical polarization withrespect to polar coordinates. Moreover, the same points on the adopted local HOP spheres ofdifferent higher-orders represent a group of higher-order VBs of locally identical polarizationbut distinct helical phase distributions. This is in contrast to the recently introduced HOPsphere which is defined with respect to globally orthonormal circular polarization states,where polarization distributions of the same order are generally not cylindrically symmetricand polarization distributions associated with the same points on HOP spheres of differentorders are not the same either. Our higher-order VBs thus form a simple and natural basis for3ffectively engineering the desired field profile and, in turn, this VB decomposition enablesan analysis of hybrid-mode excitation via VBs of various higher-orders polarization states.In addition to the match of polarization and phase distributions, an efficient concentrationof the input power on the sub-wavelength plasmonic nanoparticles is also important from apractical point of view. In this regard, the constructed incident beam needs to be tightlyfocused and the corresponding focal field is formulated exclusively in cylindrical coordinatesas a series sum of focused VBs of the associated high-order cylindrical VBs on the localHOP spheres. By doing so, we trace clearly the transfer of each polarization componentand the evolution of each higher-order polarization state on the local HOP sphere duringtight focusing. More importantly, such a focal field formulation provides an analysis tool ofthe focal field profile in higher-order polarization states and it permits a forward design ofmode-matching field profile with the input power effectively focused on nanoantennas. Wethen demonstrate that using VBs of radial polarization states of various orders is necessaryto match the polarization distributions of different hybrid modes for our example of a radialplasmonic tetramer, some of which are barely coupled with radially polarized VBs of thefundamental order used in previous works . The interaction between the tightly focusedVBs and the tetramer is efficiently simulated by the boundary element method, because thefocal fields are evaluated merely on the nanoparticles’ surfaces. Vector beams on local HOP spheres
We consider a monochromatic VB with harmonic time dependence e − i ωt , where ω is theangular frequency. The field is assumed to propagate along the z -axis and its (transverse)complex electric field amplitude at the waist plane can be written in polar coordinates ( r, φ ) as E inc ( r, φ ) = E r ( r, φ )ˆ r + E φ ( r, φ ) ˆ φ. (1)4bove E r ( r, φ ) and E φ ( r, φ ) are the radial and azimuthal field amplitudes with ˆ r and ˆ φ denoting the radial and azimuthal unit vectors, respectively. We assume that the amplitudescan be decomposed as E q ( r, φ ) = E b ( r ) P q ( r, φ ) with q ∈ { r, φ } , where E b ( r ) is a commonbeam profile shared by both field components and P q ( r, φ ) is a (complex) pupil function forphase and/or amplitude modulations of the corresponding field component. Upon choosingspecifically the pupil functions, a VB with certain spatially varying polarization state canbe engineered. Each pupil function has an azimuthal expansion P q ( r, φ ) = + ∞ (cid:88) n = −∞ c ( n ) q ( r ) e i nφ , (2)where the order n is an integer and the r -dependent expansion coefficient is c ( n ) q ( r ) = 12 π (cid:90) + π − π P q ( r, φ ) e − i nφ d φ. (3)It is convenient to introduce specific nonuniform or spatially varying polarization-statedistributions for the description of VBs. Particularly relevant ones are the unit amplitude,radially and azimuthally polarized field distributions ψ r and ψ φ obtained from Eq. (1)with E r = 1 , E φ = 0 and E r = 0 , E φ = 1 , respectively. The basis functions ψ r and ψ φ represent polarization states that are locally orthonormal . They enable us to construct,for example, the locally orthonormal right-hand circular polarization state ψ + and left-handcircular counterpart ψ − via ψ ± = ( ψ r ± i ψ φ ) / √ . We further introduce the higher-orderbasis polarization states of unit amplitude as ψ ( n ) r = e i nφ ψ r and ψ ( n ) φ = e i nφ ψ φ , (4)which possess an extra spiral phase shift nφ , as compared to their fundamental zeroth-ordercounterparts. Their linear combination forms a higher-order polarization state of the general5orm ψ ( n ) = a ( n ) r ψ ( n ) r + a ( n ) φ ψ ( n ) φ , (5)where a ( n ) r and a ( n ) φ are the complex amplitudes associated with the corresponding n th-order,radial and azimuthal basis states, respectively, which can be radius-dependent. In terms ofhigher-order polarization states defined in Eqs. (4) and (5), the VB in Eq. (1) can be rewrittenas E inc ( r, φ ) = + ∞ (cid:88) n = −∞ (cid:104) c ( n ) r ( r ) ψ ( n ) r + c ( n ) φ ( r ) ψ ( n ) φ (cid:105) E b ( r ) = + ∞ (cid:88) n = −∞ ψ ( n ) E b ( r ) , (6)with the amplitudes a ( n ) r = c ( n ) r ( r ) and a ( n ) φ = c ( n ) φ ( r ) being functions of the radius. Suchradius-dependent amplitudes permit the construction of distinctive polarization states foreach individual higher-order n at different r -values across the entire beam cross section,and summing up all higher-order VBs yields the desired field profile. On the other hand,polarization states at the same r -value are always locally identical, i.e., the ratio betweenthe radial and azimuthal field amplitudes is fixed and thus the polarization distributionson an annulus are the same with respect to polar coordinates. If the ratio c ( n ) r ( r ) /c ( n ) φ ( r ) isindependent of the radius, in particular, the vector fields across the entire transverse planeshare the same higher-order polarization state at the order n .For the polarization state of a fixed order n , we can introduce the Stokes parameters withrespect to the higher-order basis states ψ ( n ) r and ψ ( n ) φ as S ( n )0 = | a ( n ) r | + | a ( n ) φ | , (7) S ( n )1 = | a ( n ) r | − | a ( n ) φ | , (8) S ( n )2 = 2Re { [ a ( n ) r ] ∗ a ( n ) φ } , (9) S ( n )3 = 2Im { [ a ( n ) r ] ∗ a ( n ) φ } , (10)where Re and Im are the real part and imaginary part, respectively, and the asterisk denotes6igure 1: The local HOP sphere representation for VBs with local polarization states oforders n = 0 , − , (from inner to outer columns). All linear polarization states are locatedon the periphery of the S ( n )1 S ( n )2 -plane. The points + S ( n )1 and −S ( n )1 represent the n th-order,radial and azimuthal states ψ ( n ) r and ψ ( n ) φ , respectively. The point + S ( n )2 marks a linear state ψ ( n ) d with equal radial and azimuthal field components. The north pole ( + S ( n )3 ) denotes thelocally right-hand circular state ψ ( n )+ and south pole ( −S ( n )3 ) for its left-handed counterpart ψ ( n ) − . The rest of the points are occupied by elliptical states ψ ( n ) e . The instantaneous fieldsare indicated by blue arrow heads. From the temporal viewpoint, the initial phases of thestates with n = − advance by an amount of φ phase with respect to the correspondingstates of n = 0 , whereas the states of n = 2 lag behind by φ phase.the complex conjugate. The Stokes parameters in Eqs. (8)–(10) can be used for a geometricillustration of the n th-order polarization state ψ ( n ) that represents a beam with spatiallyvarying but cylindrically symmetric polarization distribution, on an adopted Poincaré sphereof radius S ( n )0 (as visualized in Fig. 1). This is in contrast to the traditional Poincaré sphererepresentation which holds for uniformly polarized beams, and it is called the local HOPsphere in this work.We remark that the basis polarization states of various orders defined in Eq. (4) only differin their instantaneous field amplitudes and, as a result, identical sets of Stokes parameters ofvarious orders represent a group of states that have the same spatial polarization distributionwith respect to the local polar coordinates but distinct (delayed) temporal behaviors. As itcan be seen in Fig. 1, polarization states at the same points on the local HOP spheres ofvarious orders are represented by polarization ellipses with locally identical orientation andshape (shown as red lines, circles, or ellipses with arrows). From a temporal point of view,on the other hand, the initial phases of n th-order states ψ ( n ) lag (for n > ) or advance (for7 < ) by an amount of | nφ | phase with respect to the corresponding th-order states ψ (0) ,as the instantaneous fields are indicated by the relative positions of the blue arrow heads onthe polarization ellipses (states of the same order are shown in the same column in Fig. 1).Here, the higher-order polarization states, Stokes parameters, and local HOP sphere providean alternative approach for describing cylindrically symmetric VBs with great simplicity andclarity by using locally identical polarization distributions, as compared with the recentlyintroduced HOP sphere in which the higher-order polarization states are defined withrespect to globally orthonormal circular polarization states. Tight focusing of vector beams
We consider the tight focusing of VBs by a high numerical aperture (NA) aplanatic lens offocal length f shown in Fig. 2(a). The objective is represented by a reference sphere shownas a light blue spherical cap. The focal field at an arbitrary point R f near the focus incylindrical coordinates ( ρ, ϕ, z ) can be obtained from the Richards–Wolf formalism byintegrating the reference field E ∞ ( φ, θ ) over the aperture, E f ( ρ, ϕ, z ) = − i k π θ max (cid:90) π (cid:90) E ∞ e i k · R f sin θ d φ d θ, (11)where k is the wave vector, k · R f = − kρ sin θ cos( φ − ϕ ) + kz cos θ (see Fig. 2), and wavenumber is k = √ k · k = n f k with n f being the refractive index after the reference sphereand k the vacuum wavenumber. The maximum angle subtended by the aperture is θ max =arcsin (NA /n f ) with NA = n f sin θ max , and we assume the beam fills the whole aperture.In refraction at the aplanatic lens, following Richards and Wolf , the reference field am-plitude vector writes as E ∞ ( θ, φ ) = f (cid:112) n i /n f √ cos θ (cid:16) E r ˆ θ + E φ ˆ φ (cid:17) where n i is the refractiveindex before the reference sphere, and the sine condition and energy conservation are used.8 a) (b) Figure 2: Tight focusing of a vector beam in an aplanatic system. (a) The incident beam E inc is mapped to the reference field E ∞ (on the light blue spherical cap with its radius beingequal to the focal length f ), the unit vectors ˆ r and ˆ θ are in the same meridional plane, and ˆ φ and ˆ θ are tangential to the reference sphere. The refractive indices before and after thereference sphere are n i and n f , respectively. The focal field E f is sought at a point R f nearthe focus, where ( ρ, ϕ, z ) form the cylindrical coordinates and ˆ ρ and ˆ ϕ are the associatedunit vectors. (b) The field components ˆ r · E ∞ and ˆ φ · E ∞ are projected into the cylindricalcoordinates ( ρ, ϕ, z ) . The unit vectors ˆ r and ˆ ρ make an angle φ − ϕ . The wave vector k measures an angle θ with respect to the z -axis and its transverse projection k ⊥ makes a φ + π angle to the x -axis.In the column vector representation in the basis [ˆ r, ˆ φ, ˆ z ] the reference field reads as ˆ r · E ∞ ( θ, φ )ˆ φ · E ∞ ( θ, φ )ˆ z · E ∞ ( θ, φ ) = L ( θ ) E r ( f sin θ, φ ) E φ ( f sin θ, φ ) , (12)where the relations r = f sin θ and ˆ θ = ˆ r cos θ + ˆ z sin θ are used, and L ( θ ) = L rr ( θ ) L rφ ( θ ) L φr ( θ ) L φφ ( θ ) L zr ( θ ) L zφ ( θ ) = f (cid:114) n i n f √ cos θ cos θ
00 1sin θ (13)can be understood as the matrix representation for the refraction at the aplanatic lens. Thematrix element L sq ( θ ) with s ∈ { r, φ, z } and q ∈ { r, φ } denotes the field amplitude transferfrom the q -component of the incident VB to the s -component of the reference field.9o perform the integration in Eq. (11), the reference field needs to be expanded in thesame basis as the focal field. A convenient choice is the basis [ ˆ ρ, ˆ ϕ, ˆ z ] , in which the columnvector form of Eq. (11) becomes ˆ ρ · E f ( ρ, ϕ, z )ˆ ϕ · E f ( ρ, ϕ, z )ˆ z · E f ( ρ, ϕ, z ) = − i k π θ max (cid:90) π (cid:90) ˆ ρ · E ∞ ( θ, φ )ˆ ϕ · E ∞ ( θ, φ )ˆ z · E ∞ ( θ, φ ) e i k · R f sin θ d φ d θ. (14)Above the vector form of E ∞ ( θ, φ ) in the basis [ ˆ ρ, ˆ ϕ, ˆ z ] is connected to its representation inthe basis [ˆ r, ˆ φ, ˆ z ] through ˆ ρ · E ∞ ( θ, φ )ˆ ϕ · E ∞ ( θ, φ )ˆ z · E ∞ ( θ, φ ) = T ( φ, ϕ ) ˆ r · E ∞ ( θ, φ )ˆ φ · E ∞ ( θ, φ )ˆ z · E ∞ ( θ, φ ) (15)with the connection matrix T ( φ, ϕ ) = T ρr T ρφ T ρz T ϕr T ϕφ T ϕz T zr T zφ T zz = cos( φ − ϕ ) − sin( φ − ϕ ) 0sin( φ − ϕ ) cos( φ − ϕ ) 00 0 1 , (16)obtained by recognizing, from Fig. 2, the relations of field components ˆ ρ · E ∞ = ˆ r · E ∞ cos( φ − ϕ ) − ˆ φ · E ∞ sin( φ − ϕ ) (17)and ˆ ϕ · E ∞ = ˆ r · E ∞ sin( φ − ϕ ) + ˆ φ · E ∞ cos( φ − ϕ ) . (18)The matrix element T ps ( φ, ϕ ) with p ∈ { ρ, ϕ, z } and s ∈ { r, φ, z } characterizes the focalfield’s p -component that arises from the s -component of the reference field.10ubstituting Eq. (12) into Eq. (15), as well as using Eqs. (1) and (6), the focal field invector form in the basis [ ˆ ρ, ˆ ϕ, ˆ z ] writes as ˆ ρ · E f ( ρ, ϕ, z )ˆ ϕ · E f ( ρ, ϕ, z )ˆ z · E f ( ρ, ϕ, z ) = − i k π θ max (cid:90) π (cid:90) T ( φ, ϕ ) L ( θ ) + ∞ (cid:88) n = −∞ c ( n ) r ( f sin θ ) c ( n ) φ ( f sin θ ) e i nφ × E b ( f sin θ ) e − i kρ sin θ cos( φ − ϕ ) e i kz cos θ sin θ d φ d θ. (19)Performing the integration of the φ -dependent portion at a given order n , we can write − i k π π (cid:90) T ( φ, ϕ ) e i nφ e − i kρ sin θ cos( φ − ϕ ) d φ = e i nϕ ( − i) n k ( J n − − J n +1 ) − i( J n − + J n +1 ) 0i( J n − + J n +1 ) ( J n − − J n +1 ) 00 0 2 J n = e i nϕ Θ ( n ) ρr Θ ( n ) ρφ Θ ( n ) ρz Θ ( n ) ϕr Θ ( n ) ϕφ Θ ( n ) ϕz Θ ( n ) zr Θ ( n ) zφ Θ ( n ) zz = e i nϕ Θ ( n ) , (20)where J n = J n ( kρ sin θ ) denotes the n th order Bessel function with argument kρ sin θ forbrevity, the n th-order matrix Θ ( n ) ( θ, ρ, ϕ ) and its elements are introduced such that theabove equation holds, and we have used the integral representation of the Bessel function (cid:90) π e i nα e − i x cos α d α = (cid:90) π e i nα e i x cos( α + π ) d α = 2 π ( − i) n J n ( x ) . (21)Expressing the remaining integration over θ for a given order n compactly, we introduce I ( n ) p ( ρ, z ) = θ max (cid:90) (cid:88) s,q Θ ( n ) ps ( θ, ρ, ϕ ) L sq ( θ ) c ( n ) q ( f sin θ ) E b ( f sin θ ) e i kz cos θ sin θ d θ, (22)with the subscripts p ∈ { ρ, ϕ, z } , s ∈ { r, φ, z } , and q ∈ { r, φ } . Overall, the focal field11ecomes, E f ( ρ, ϕ, z ) = (cid:88) n E ( n ) f = (cid:88) n [ I ( n ) ρ ( ρ, z ) ψ ( n ) ρ + I ( n ) ϕ ( ρ, z ) ψ ( n ) ϕ + I ( n ) z ( ρ, z ) e i nϕ ˆ z ] , (23)where the n th-order transverse field is expanded on the higher-order basis states ψ ( n ) ρ = e i nϕ ψ ρ and ψ ( n ) ϕ = e i nϕ ψ ϕ with ψ ρ and ψ ϕ being the unit-amplitude, radially and azimuthallypolarized field distributions with respect to the cylindrical coordinates ( ρ, ϕ, z ) . The tightlyfocused field also acquires a longitudinal component that has the same helical phase variationas the transverse focal field. Compared with the fundamental order longitudinal field (so-called optical needle ), the higher-order ones may have profound impacts in applicationssuch as second-harmonic generation and single emitter probing in microscopy .For hybrid-mode excitation of in-plane nanoantennas, we can consider only the transversefocal field whose spatially varying polarization distributions can be studied with the Stokesparameters introduced in Eqs. (7)–(10), where the associated complex amplitudes are a ( n ) ρ = I ( n ) ρ ( ρ, z ) and a ( n ) ϕ = I ( n ) ϕ ( ρ, z ) . In addition, the incident field at the n th-order polarizationstate [in Eq. (6)] is converted to the focal field at the polarization state of the same order n [inEq. (23)]. This implies that the incident VB and the transverse part of the focal field have thesame amount of helical phase or, in other words, the same orbital angular momentum. But,the transverse focal field generally does not preserve the same local polarization distributionsas the incident VB. This results from energy exchanges between the radial and azimuthal fieldcomponents in the focusing process due to the presence of nonzero elements Θ ( n ) ϕr and Θ ( n ) ρφ for n (cid:54) = 0 . Furthermore, the polarization distributions are both ρ - and z -dependent accordingto Eq. (22). Consequently, the transverse field near the focus generally does not have a pureradial or azimuthal polarization distribution except for the fundamental zeroth-order.A typical example of paraxial VBs is a cylindrically polarized Laguerre-Gaussian (CPLG)beam , where the common beam profile E b ( r ) = exp − τ / is a Gaussian with τ = √ r/w and w being the radius of the beam waist. For this beam, the n th-order expansion coefficient12n Eq. (2) takes the following explicit form: c ( n ) q ( r ) = A ( n ) q τ n ∓ L n ∓ m ( τ ) , (24)with A ( n ) q being the amplitude of the corresponding field component, L n ∓ m the generalizedLaguerre polynomials, m the radial index, and n ∓ the azimuthal index. For the fundamen-tal order when n = 0 , the plus sign should be used in the power and azimuthal order so that c (0) q ( r ) ∝ τ L m ( τ ) yields a null field on the optical axis where the phase has a singularity.We use the explicit form of the expansion coefficients in Eq. (24) and choose the “ + ” signfor all simulations in this paper. -500 0 500 -5000500 x [ ] y [] (a) -2 -1 0 1 2 -2-1012 x [ ] y [] (b) -2 -1 0 1 2 -2-1012 x [ ] y [] (c) -500 0 500 -5000500 x [ ] y [] (d) -2 -1 0 1 2 -2-1012 x [ ] y [] (e) -2 -1 0 1 2 -2-1012 x [ ] y [] (f) Figure 3: Amplitude and polarization distributions of ψ (3) r and its focused VBs. Fieldamplitudes are shown in greyscale colormaps for (a) the incident radial | E r | , (b) the focusedradial | E ρ | , (c) the focused azimuthal | E ϕ | , and (f) the focused longitudinal | E z | components,respectively. The polarization distributions are shown in (d) for the paraxial and (e) thefocused transverse fields, where the phases of the radial components are indicated in blueto red colormaps. The blue or red ones imply locally in-phase instantaneous fields and thelight green ones indicate out-of-phase cases.Figure 3 illustrates the field amplitude and polarization distributions of a ψ (3) r beam and13ts focused counterpart. In detail, the incident VB has a waist radius of w = 0 . mm atthe wavelength λ = 1060 nm, the radial index is m = 0 , the azimuthal index is n + 1 = 4 ,and the azimuthal field amplitude A (3) φ is set to zero. In addition, the objective has an NA of . and the focal length is f = 1 mm. The radial field amplitude | E r | of the ψ (3) r beamis shown in greyscale in Fig. 3(a), and its polarization distribution is shown in Fig. 3(d) onwhich the phase distribution of the radial field arg( E r ) is also superimposed in a blue to redcolormap. Radial and azimuthal field amplitudes | E ρ | and | E ϕ | in the focal plane generatedin tight focusing are shown in Fig. 3(b) and (c), respectively. As indicated by the greyscalecolormaps, the azimuthal and radial field amplitudes depend differently on the radius and,for some radii, the azimuthal amplitudes even exceed its radial counterparts. Consequently,the transverse focal field is in general polarization state as shown in Fig. 3(e). Nevertheless,the order of polarization states remains unchanged throughout the focusing process. In otherwords, the field in Fig. 3(d) shares the same higher-order radial polarization state that isrepresented at the point + S (3)1 on the local HOP sphere. By contrast, the polarization statesof the transverse focal fields in Fig. 3(e) spread over the local HOP sphere of the same order.As the radius ρ increases, the point on the local HOP sphere representing the polarizationstate of the transverse focal field shifts along the meridian from the north pole + S (3)3 (thethird-order right-hand circular polarization) to the south pole −S (3)3 (the third-order left-hand circular polarization) via the point −S (3)1 (the third-order azimuthal polarization), andthen back to the north pole + S (3)3 via the point + S (3)1 (the third-order radial polarization).At last, a significant longitudinal field in the focal region is also shown in Fig. 3(f). Such astrong longitudinal component twists the major axes of the polarization ellipses out of thefocal plane onto a Möbius strip at a given r -value.14 fficient hybrid mode excitation from plasmonic nanoan-tennas In this section, we proceed to show that focused VBs can efficiently excite distinct hybridmodes in plasmonic nanoantennas by providing not only matching polarization distributionsbut also strong incident field on the nanoparticles. Compared with the previous worksthat used cylindrical VBs (of the fundamental order when analyzed in terms of the higher-order polarization states) , here we demonstrate the importance of using VBs in higher-order polarization states to interact with a number of hybrid modes. As an example, wetheoretically study light scattering from a plasmonic tetramer consisting of four identical,radially oriented gold nanorods. Each individual nanorod supports a dipole-like localizedsurface plasmon mode that is associated with the rod’s long axis and can be excited by amatching, linearly polarized plane wave or Gaussian beam. In the radial tetramer, a numberof hybridized modes of cylindrically symmetric polarization distributions emerge from near-field coupling among the dipolar modes. We compare the excitation efficiencies of varioushybrid modes in the tetramer when illuminated by paraxial VBs on the fundamental zeroth-order, first-order, and second-order radial polarization states, as well as the correspondingfocused beams. The excitation efficiencies are characterized by the total powers scatteredfrom the tetramer, which are numerically calculated by the boundary element method .We use the formulations given in the previous sections to calculate the focal electric fields.The associated magnetic fields in the focal region are readily obtained by replacing E f with H f and E ∞ with H ∞ = (1 /Z f )( k /k ) × E ∞ in Eq. (11) , where Z f is the wave impedanceafter the reference sphere. From a computational point of view, the boundary elementmethod is very efficient for modeling the interaction between tightly focused beams andplasmonic nanoantennas, since the excitation focal fields are evaluated only on the surfacesof the nanoparticles. In all simulations, the same parameters as in Fig. 3 are used. Inaddition, the middle plane of the tetramer coincides with the focal plane where z = 0 . Each15onstituent nanorod has a width of nm, thickness of nm, and length of nm. Eachpair of opposite nanorods has an end-to-end separation of nm, and the smallest distanceof every neighboring nanorods are then . nm between rounded corners. The nanorodsare embedded in a medium of effective refractive index n f = 1 . , and the refractive indexof gold nanorods is taken from the tabulated data . r [µm] | E r | [ V / m ] (a)
50 100 150 20010 -13 -10 -7 -4 -1 r [nm] | E r | [ V / m ] (b) [µm] | E | [ V / m ] (c)
50 100 150 20010 [nm] | E | [ V / m ] (d) Figure 4: The radial field amplitude distributions of the paraxial ψ (0) r , ψ (1) r , and ψ (2) r VBsare shown in (a) and (b), and the focused counterparts denoted by symbols E (0) f , E (1) f , and E (2) f , respectively, are displayed in (c) and (d). The vertical dashed lines in (b) and (d) markthe nanorod’s center.The radial field amplitude distributions of the paraxial ψ (0) r , ψ (1) r , and ψ (2) r VBs are shownin Fig. 4(a) for a radius range from to µ m, as well as in Fig. 4(b) for the radius rangearound the nanorods with r ∈ [0 , nm. Each paraxial VB has a unit peak-field amplitudeand its energy is mainly distributed around − µ m away from the optical axis. Thefield amplitudes on the nanorods [as the nanorod’s center is indicated by the vertical dashedline in Fig. 4(b)] become very weak; they drop down to ∼ − , − and − V/m forthe paraxial ψ (0) r , ψ (1) r , and ψ (2) r VBs, respectively. The focused radial fields of the paraxial16 (0) r , ψ (1) r , and ψ (2) r VBs are plotted in Fig. 4(c) and (d) for the radius ranges of [0 , µ mand [0 , nm, separately. It is seen that the focused VB’s energy concentrates down tothe wavelength scale, and the field amplitudes around the nanorods are ∼ V/m whichare several orders of magnitude higher than those of paraxial counterparts. -0.2 0 0.2-0.200.2 x [µm] y [ µ m ] (a) -0.2 0 0.2-0.200.2 x [µm] y [ µ m ] (b) -0.2 0 0.2-0.200.2 x [µm] y [ µ m ] (c) -0.2 0 0.2-0.200.2 x [µm] y [ µ m ] (d) -0.2 0 0.2-0.200.2 x [µm] y [ µ m ] (e) -0.2 0 0.2-0.200.2 x [µm] y [ µ m ] (f)
800 900 100001234 x1.0E-17 x1.4E-24 x5.3E-11 monomer wavelength [nm] P sc a [ n W ] (g)
800 900 100001234 x1.5E1 x5.4E-1 x5.6E-1 monomer wavelength [nm] P sc a [ n W ] (h) Figure 5: The excitation of hybrid modes in the radial tetramer. The polarization distri-butions around the tetramer (solid black lines) in the focal plane are shown in (a)–(c) forthe paraxial VBs and (d)–(f) for the focused VBs. The phases of the radial fields are visu-alized in blue to red colormaps. The radial field amplitudes on nanorods’ centers (crossedby black dashed curves) are . × − in (a), × − in (b), . × − in (c), in (d), in (e), and V/m in (f). The total powers scattered from the tetramer are plottedin the blue, yellow, and red curves in (g) when illuminated by the paraxial ψ (0) r , ψ (1) r , and ψ (2) r VBs of unit-amplitude peak field or (h) the corresponding focused E (0) f , E (1) f , and E (2) f VBs, respectively. The power spectra of a single independent nanorod (monomer) is plottedas a reference in the black dashed curve. The absolute power values are obtainable whenmultiplied by the factors near each spectral peak.The polarization distributions of the paraxial ψ (0) r , ψ (1) r , and ψ (2) r VBs are shown inFig. 5(a)–(c), and the phase distributions are visualized in blue to red colormaps. Theparaxial VBs are in the polarization states represented by the points + S (0)1 , + S (1)1 , and17 S (2)1 , respectively, on the local HOP spheres of the corresponding orders, and their electricfields are all radially polarized that match individually with the localized surface plasmonsalong the nanorod’s long axes. The instantaneous electric fields of the fundamental ψ (0) r VBon all nanorods are in phase (drawn as blue arrows). It yields an excitation of a hybrid modethat features two pairs of in-phase, antibonding localized surface plasmons with a blue-shiftedresonant wavelength at ∼ nm [see the blue solid curve in Fig. 5(g)], as compared to theplasmon resonance of an independent nanorod at ∼ nm (the black dashed curve). For thefirst-order ψ (1) r VB, the instantaneous fields on opposite nanorods are π out-of-phase whichare visualized by either blue to light green arrows or dark green to yellow arrows. The hybridmode excited by the ψ (1) r VB has a red-shifted resonance at ∼ nm [the yellow curve inFig. 5(g)], arising from a bonding localized surface plasmon that rotates, as a function oftime, between two pairs of opposite nanorods. In the case of the second-order ψ (2) r VB, theneighboring nanorods experience π out-of-phase instantaneous fields, and two pairs of out-of-phase, antibonding localized surface plasmons emerge and contribute to a hybrid mode whoseresonance is red-shifted to ∼ nm [the red curve in Fig. 5(g)]. The paraxial VBs, whentheir polarization distributions are appropriately tuned, are able to excite various hybridmodes that possess distinct spectra with well-separated resonant wavelengths. However, thecorresponding excitation efficiencies are extremely low, as indicated by the multiplicationfactors ( . × − , . × − , and . × − on the blue, yellow, and red spectralcurves, respectively) for obtaining the absolute values of the total scattered power. Thelow efficiencies are attributed to the weak incident field amplitudes effectively exerted onthe nanorods, regardless of the perfect matches of polarization distributions between theincident fields and the localized surface plasmons of the hybrid modes. In detail, the fieldamplitudes on the nanorods’ centers are only . × − , . × − , and . × − V/m[see the factors around the dashed circles in Fig. 5(a)–(c)] for the paraxial ψ (0) r , ψ (1) r , and ψ (2) r VBs, separately.The situation is significantly improved when the paraxial VBs are tightly focused. Fig-18re 5(d)–(f) display the polarization distributions of the transverse fields and the phasedistributions of the radial fields of the E (0) f , E (1) f , and E (2) f VBs that are tightly focused fromthe paraxial ψ (0) r , ψ (1) r , and ψ (2) r VBs, respectively. The factors close to the dashed circlesmark the radial field amplitudes on nanorods’ centers, which are , , and V/m forthe focused E (0) f , E (1) f , and E (2) f VBs, separately. The strong, effective radial fields of thefocused VBs significantly elevate the excitation efficiencies of the associated hybrid modesby approximately , , and orders of magnitude, as seen from Fig. 5(g) and (h).The focused E (0) f VB’s transverse field is in the polarization state that is representedby the same + S (0)1 point on the local HOP sphere as its paraxial counterpart except forfield concentration and enhancement (i.e., the local HOP sphere’s radius S (0)0 increases) onthe nanoscale tetramer. Therefore, it interacts with the same hybrid mode as its paraxialcounterpart except for yielding significantly high efficiency. For both the focused E (1) f and E (2) f VBs, the azimuthal field components emerge from the nonzero I (1) ϕ ( ρ, and I (2) ϕ ( ρ, in Eq. (23). Furthermore, the ratios I (1) ρ ( ρ, /I (1) ϕ ( ρ, and I (2) ρ ( ρ, /I (2) ϕ ( ρ, are radius-dependent. Therefore, the polarization states of their transverse fields become ellipticallyor even circularly polarized [see the polarization ellipses in Fig. 5(e) and (f)] which canbe represented by points near the north poles + S (1)3 and + S (2)3 on the corresponding localHOP spheres. Since the azimuthal fields barely couple to the hybrid modes whose localizedsurface plasmons have dominant radial polarizations, they can be ignored here. On theother hand, the remaining transverse fields are in radial polarization states at the + S (1)1 , and + S (2)1 points on the local HOP spheres with much larger radii S (1)0 and S (2)0 , i.e., enhancedfield amplitudes on the tetramer, than those of the paraxial counterparts. This explainsthe increases in excitation efficiencies for the corresponding hybrid modes. It is also worthnoticing that there is actually no phase singularity for the central field of the focused E (1) f VB and a nonzero field on the optical axis [see the yellow curve in Fig. 4(d)] manifests itselfduring the tight focusing. It also implies that we could have chosen the “ − ” sign in Eq. (24)for the paraxial ψ (1) r VB. Even in that particular choice for the paraxial ψ (1) r VB where the19adial field would have a peak in the center, the hybrid-mode excitation efficiency wouldbe still a few orders lower than in the situation of using the focused VB. Therefore, tightfocusing of the paraxial VBs is needed in general for obtaining high hybrid-mode excitationefficiencies.Although we have only demonstrated the efficient excitation of hybrid modes in a radialplasmonic tetramer by using tightly focused VBs of various higher-order radial polarizationstates, the developed framework for shaping and analyzing the focused VB’s higher-orderpolarization states is readily extended to cases involving hybrid modes in other plasmonictetramers or other oligomers aggregated of nanorods or nanoparticles of other shapes. Forinstance, if the constituent nanoparticles are of disk shape, focused VBs of higher-ordercircular polarization states can be of paramount importance.
Conclusions
We investigated a general approach to engineer tightly focused VBs that can efficientlyand selectively couple with various hybrid modes in plasmonic nanoantennas. We usedhigher-order radially and azimuthally polarized VBs as a natural basis to construct theparaxial VB of desired field profile, which was then tightly focused to effectively deliver theinput power on a sub-wavelength nanoantenna where the focal field is still effective mode-matching. The desired field profile was obtained in the paraxial VB by superposing higher-order VBs and tuning the radius-dependent expansion coefficients. We adopted the conceptof higher-order VBs on local HOP spheres, whose polarization distributions are cylindricallysymmetric that have locally identical polarizations but distinctive instantaneous electricfields as a manifestation of the helical phase distributions. This description provides a clearunderstanding of the polarization transfers from the incident paraxial VBs into the focusedones, and thus the modification in the field profile was traced during focusing. Furthermore,it permits a backward optimization or inverse design to alleviate unnecessary polarizations20n the focal field. We theoretically examined the efficient excitation of hybrid modes in aradial plasmonic tetramer by radially polarized paraxial VBs of various higher-orders andtheir focused counterparts. We explained, via our developed VB analysis in higher-orderpolarization states, that the corresponding focused VB shares a significant field component inthe same polarization distribution as the paraxial one which permits a much higher efficiencyfor the associated hybrid mode. This example demonstrates the importance of higher-orderVBs for the efficient, selective hybrid-mode excitation and it also indicates the great potentialof tightly focused vector beams for tailoring light scattering from plasmonic nanoantennasthrough controlled hybrid-mode excitation.
Acknowledgments
The authors appreciate fruitful discussions with Prof. Marco Ornigotti. This work getsthe financial support from the Academy of Finland (287651) and the Flagship of PhotonicsResearch and Innovation (PREIN) (320165, 320166).
Disclosures
The authors declare no conflict of interest.
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