Universal Quasi-Static Limit for Plasmon-Enhanced Optical Chirality
Marco Finazzi, Paolo Biagioni, Michele Celebrano, Lamberto Duó
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Universal Quasi-Static Limit for Plasmon-Enhanced OpticalChirality
Marco Finazzi, ∗ Paolo Biagioni, Michele Celebrano, and Lamberto Du`o
Dipartimento di Fisica and CNISM, Politecnico di Milano,Piazza Leonardo da Vinci 32, 20133 Milano, Italy (Dated: July 7, 2018)
Abstract
We discuss the possibility of enhancing the chiroptical response from molecules uniformly dis-tributed around nanostructures that sustain localized plasmon resonances. We demonstrate thatthe average optical chirality in the near field of any plasmonic nanostrucure cannot be signifi-cantly higher than that in a plane wave. This conclusion stems from the quasi-static nature ofthe nanoparticle-enhanced electromagnetic fields and from the fact that, at optical frequencies, themagnetic response of matter is much weaker than the electric one.
PACS numbers: 42.25.Ja,33.55.+b, 78.20.Ek ∗ Electronic address: marco.fi[email protected] )b) c)d)
FIG. 1: (Color online) Uniform distribution of chiral molecules (red dots) on the surface [(a) and(b)] or in the volume [(c) and (d)] surrounding a nanoparticle suspended in a fluid [(a) and (c)] ordeposited on a substrate [(b) and (d)].
We will evaluate the realistic case consisting in a homogeneous distribution of chiralmolecules in a domain D encircling a nanoparticle. D might coincide with the particlesurface in the case the latter has been functionalized with organic surfactants binding themolecules, as in Figs. 1(a) and 1(b), or with the volume around the particle, as in Figs. 1(c)and 1(d) describing the situation where the molecules are suspended in the surroundingliquid.The signal-to-noise ratio in a dichroic absorption experiment is proportional to g I [23], with I being the intensity of the light which is modulated between the two op-posite circular polarizations and g the dissymmetry factor. The latter is defined as g = 2 ( A + − A − ) / ( A + + A − ), where A ± is the absorption rate for the two electromag-netic fields interchanged by parity. Tang and Cohen [7] demonstrated that, for randomlyoriented molecules subject to electric and magnetic dipole transitions at frequency ω , g = − (cid:18) G ′′ α ′′ (cid:19) ω CU E , (1)where C and U E are the time-averaged optical chirality and electric energy density, respec-tively. The quantity in round brackets is related only to the physical properties of the chiralmolecules and depends on α ′′ and on G ′′ , the imaginary part of the electric polarizabilityand the isotropic mixed electric-magnetic dipole polarizability, respectively.Since we are interested in evaluating the dichroic signal from the ensemble of chiralmolecules, we need to determine C by averaging over the homogeneous domain D thatcontains the molecules. We consider harmonic electromagnetic fields, which can be expressed3s follows: E ( r , t ) = ˆ E ( r ) e − iωt , B ( r , t ) = ˆ B ( r ) e − iωt , (2)where ˆ E ( r ) and ˆ B ( r ) are complex vectors: only the real parts of E ( r , t ) and B ( r , t ) describethe physical electromagnetic fields. Following Ref. 7 we can then write C = − V D ǫω (cid:20)Z D ˆ E ∗ ( r ) · ˆ B ( r ) d r (cid:21) , (3) ǫ being the permittivity of the medium surrounding the particle and V D standing for thearea [see Figs. 1(a) and 1(b)] or volume [Figs. 1(c) and 1(d)] of the domain D .For the same reason, in Eq. (1) we need to consider the D -averaged U E value, equal to U E = 1 V D ǫ Z D | ˆ E ( r ) | d r . (4)By applying the Cauchy-Schwarz inequality it is possible to establish a limiting upper valuefor | g | : | g | ≤ (cid:12)(cid:12)(cid:12)(cid:12) G ′′ α ′′ (cid:12)(cid:12)(cid:12)(cid:12) ǫ V D (cid:12)(cid:12)(cid:12)R D ˆ E ∗ ( r ) · ˆ B ( r ) d r (cid:12)(cid:12)(cid:12) U E ≤ (cid:12)(cid:12)(cid:12)(cid:12) G ′′ α ′′ (cid:12)(cid:12)(cid:12)(cid:12) c h V D ǫ R D | ˆ E ( r ) | d r i h V D µ R D | ˆ B ( r ) | d r i U E = (cid:12)(cid:12)(cid:12)(cid:12) G ′′ α ′′ (cid:12)(cid:12)(cid:12)(cid:12) c r U B U E = | g max | . (5)In this inequality µ and c are the permeability and the speed of light in the medium hostingthe chiral molecules, while U B is the total time- and space-averaged magnetic energy densityin D : U B = 1 V D µ Z D | ˆ B ( r ) | d r . (6)In a plane wave U B /U E = 1, but more favorable field geometries indeed exist. Thestrategy devised by Tang and Cohen to maximize the g value and obtain superchiral fieldscharacterized by U B ≫ U E consists in choosing the integration domain D around a nodalsurface of the electric field ˆ E ( r ) in a stationary wave, hence minimizing the value of U E with respect to U B [7]. In practice this is obtained by reflecting circularly polarized light atnormal incidence off a mirror with reflectivity close to unity [8].A fist possible strategy to translate Tang and Cohen’s approach into the plasmonic realmwould be to replace the mirror with a metal nanoparticle. The electric field ˆ E p generated4y the charges induced in the particle by the illuminating field ˆ E can be rather accuratelydescribed within the so-called quasi static approximation [24], which neglects retardationeffects. In this case the electric field ˆ E p ( r ) in the domain D can be expressed as the gradientof a time-varying electrical potential satisfying the Laplace equation, therefore fulfilling thecondition ∇ × ˆ E p ( r ) = 0, while the curl of the illumination field is given by ∇ × ˆ E ( r ) = ∂ ˆ B ( r ) /∂t . As a consequence, ˆ E p and ˆ E cannot cancel each other on a connected surfacewrapping the particle because, according to the Kelvin-Stokes theorem, they must havedifferent line integrals over any closed loop [25].Note that this argument does not exclude that ˆ E ( r ) = ˆ E p ( r ) + ˆ E ( r ) ≈ locally superchiralat selected positions in the proximity of a nanoparticle [22]. However, the impossibilityto cancel the impinging electric field with the one scattered by the particle leads to theconclusion that the only way to realize a large average optical chirality in the near field of anoptical nanoantenna would be to engineer geometries able to generate much larger magneticthan electric near fields. In this way one might fulfill the required condition U B ≫ U E (with U E = 0) to obtain | g max | values significantly higher than those expected for dichroicexperiments performed with plane-wave illumination. Since the magnetic response of a nonmagnetic nanoparticle can be sizeable only when the wavelength of the illuminating light istuned at one of the particle resonances [26], one should look for particle geometries sustainingquasi-normal modes characterized by large magnetic multipole moments as compared withthe corresponding electric ones. Since plasmon resonances are characterized by high local-field intensity enhancements, in the following we will neglect any interference between theparticle quasi-static near field and the illuminating light.With plane-wave illumination at wavelength λ , the probability of inducing an electric ormagnetic multiplole moment in a particle of size d < λ located at r = 0 rapidly falls off withmultipole order [25]. For nonmagnetic particles (i.e. particles with vanishing magnetization),the leading-order (in d/λ ) contributions are given by the following expressions [3]: p i = X j,k α ij ˆ E j (0) + 13 A ijk ∂ ˆ E j ∂x k (0) + G ij ˆ B j (0) + . . . (7a) m i = X j γ ij ˆ B j (0) + G ∗ ij ˆ E j (0) + . . . (7b)5 ij = X k A ∗ ijk ˆ E k (0) + . . . (7c)In these equations, p i , m i and q ij are the electric dipole, magnetic dipole and electricquadrupole, respectively, while α ij and γ ij are the particle polarizability and magnetiz-ability tensors. G ij is the mixed electric-magnetic dipole polarizability of the particle, while A ijk is a third-order tensor that does not contribute to the particle magnetic moment.At this point, it is worth to remind that, in atoms and molecules, α ij , γ ij ,and G ij followa well-defined hierarchy: | α ij /G i ′ j ′ | ≈ λ/d m and | α ij /γ i ′ j ′ | ≈ ( λ/d m ) , with d m being themolecule (or atom) size [26]. Nanoparticles, however, do not necessarily need to followthis scaling rule. Magnetic multipoles comparatively larger than the corresponding electricones might in principle be obtained with a proper choice of the nanoantenna geometry andmaterial. However, in metals, this is possible only in the small skin-depth limit associatedwith the condition κ ≫ λ/ (2 πd ) [26], where κ is the imaginary part of the complex refractionindex of the particle material, λ/ (2 πd ) typically being of the order of unity for Ag or Aunanoparticles sustaining plasmonic resonances in the visible. Unfortunately, such conditionis not really met for these materials, where 2 . κ . ≤ λ ≤
800 nm [27]. Wecan thus exclude that a space-integrated optical chirality significantly larger than that of acircularly polarized plane wave could ever be expected in the near field of any plasmonicnanostructure displaying a resonance in the visible region of the electromagnetic spectrum.To provide further evidence to our thesis and reinforce our conclusions, we have per-formed numerical simulations with the finite-difference time-domain method [28] of theelectromagnetic fields in the proximity of Au nanospheres, which are widely employed inmany applications of plasmonics and nanosensing. Following the above discussion, sphericalnoble-metal nanoparticles are ill-suited for solution-phase measurements of the chiropticalproperties of molecules, as already pointed out in Ref. 14. This is confirmed by Fig. 2, whichreports the integrated | g/g pw | value calculated around an Au particles with diameter D = 50and 200 nm as a function of the excitation wavelength λ , g pw being the dissymmetry factorof the same molecules as measured with circularly polarized plane waves, i.e. without thenanoparticle. Figure 2 also displays | g max /g pw | and the electric field intensity enhancementaveraged over the spherical surface of the particles [29]. Simulations thus clearly supportthe conclusion U B < U E and, consequently, | g | < | g max | < | g pw | , also for the sphere with D = 200 nm, for which retardation effects cannot be completely neglected.6 Wavelength (nm)
500 90080070060000 I n t en s i t y enha c e m en t I n t en s i t y enha c e m en t kk Intensityenhancement g g max pw / g g / pw FIG. 2: (Color online) Spectral dependence of the g/g pw (full lines) and g max /g pw (dash-dottedlines) for Au nanospheres with diameter D = 50 nm (a) and 200 nm (b). The respective electricfield intensity enhancement spectra are also reported (dotted lines). The insets display the spacedistribution of the ratio ( C − C pw ) /C pw ( C pw being the optical chirality of the plane wave) in thesphere symmetry plane parallel to the light wavevector k , evaluated at the maximum near fieldenhancement. The black bar in the insets represents a length of 100 nm. The same result has been obtained for a different geometry (see Fig. 3a), namely for agold rod with spherical terminations (cross section diameter D = 50 nm, total length L =160 nm). Such a geometry leads to a lowest-order resonance located at λ ≈
710 nm and to anelectric field intensity enhancement of about a factor 85. This higher field-enhancement valuecompared to the one obtained for the Au nanospheres stems from the fact that the nanorodresonance is located in a spectral region where interband transitions in gold do not occurand the material is thus characterized by much lower losses. Nevertheless, the rod geometryis equally deceiving in terms of optical chirality, as expected from the previous discussion.Noteworthy, a similar result ( | g | < | g max | < | g pw | ) is also obtained for a longer rod ( L =580 nm, D = 50 nm), which displays a third order resonance of the guided plasmonic modein the same spectral range (see Fig. 3b). This suggests that the conclusions derived abovecan, to some extent, be prolonged to particles with a larger size than the light wavelength.7he optical behavior of the nanostructures described above is dominated by electricdipole (or multipole) resonances. Split-ring resonators, on the other side, are known tosustain circulating currents with a large associated magnetic dipole [30]. However, it is alsoknown that their large kinetic inductance prevents such resonances from entering the visiblespectral range [31]. Recently, Feichtner et al . demonstrated that a novel antenna geometry,merging together the characteristics of a dipole and of a split-ring antenna, is able to push thesplit-ring resonance below 700 nm wavelength [32]. This geometry, therefore, represents aninteresting benchmark to test the limitations outlined above for plasmon-enhanced chiralityalso in the case non-dipolar resonances. Neverthless, this structure is not able to deliveran averaged optically chirality higher than the one associated to a plane wave, showing abehavior qualitatively similar to the one displayed by the previously discussed geometries(see Fig. 3c).We would like to conclude by noting that, for all the considered geometries, the space-averaged optical chirality of the fields surrounding the particles shows a pronounced mini-mum at the plasmon resonance. This is a consequence of a large particle polarizability andenhanced electric fields, to be compared to the much smaller magnetic fields resulting froma weak particle magnetizability.In summary, we have demonstrated that the difficulty of realizing nanostructures capableof generating near fields with a high average optical chirality is related to fundamental phys-ical limitations to all subwavelength noble-metal particles. This limitation is a consequenceof both the quasi-static nature of the nanoparticle-enhanced electromagnetic fields and ofthe fact that the magnetic response of matter cannot be much higher than the electric oneat optical frequencies. This conclusion suggests a twofold strategy to exploit the locallyenhanced fields produced by nanostructures to perform high-sensitivity experiments aimingat characterizing the chiroptical properties of molecules. By recalling that the signal-to-noise ratio in a dichroism experiment depends on g I ( I being the light intensity), the firstpossibility would be to selectively place the molecules to be investigated in specific spotswhere properly engineered particles could concentrate superchiral light ( g ≫ g ≃
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