Fundamental Limits to the Coupling between Light and 2D Polaritons by Small Scatterers SUPPORTING INFORMATION

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1 Fundamental Limits to the Coupling between Light and D Polaritons by Small Scatterers SUPPORTING INFORMATION Eduardo J. C. Dias and F. Javier García de Abajo ICFO-Institut de Ciéncies Fotòniques The Barcelona Institute of Science and Technology Castelldefels (Barcelona Spain ICREA-Institució Catalana de Recerca i Estudis Avançats Passeig Lluís Companys Barcelona Spain (Dated: March 8 09 We provide an alternative derivation of the power scattered by a dipolar particle based on the Poynting vector numerical simulations for the polarizability of gold and silver nanoparticles as a function of their size and an extension of the results of the main text to include retardation. Contents S. Poynting-Vector-Based Calculation of the Scattered Power S. Fresnel Coefficients 3 S3. Results with Retardation 4 S4. Influence of Size and Shape of the Scatterer 5 S5. Polarizability of a Circular Hole 7 References 7 S. POYNTING-VECTOR-BASED CALCULATION OF THE SCATTERED POWER In this section we provide an alternative derivation of eq 5 of the main text based on the Poynting vector for which we perform fully electrodynamic calculations because we need the magnetic field that otherwise vanishes in the quasistatic limit. We focus on the system depicted in Figure b with a dipole p placed right above the z = 0 planar interface that separates regions of permittivities and ɛ m. For reference we divide space into the three regions I II and III defined by the conditions z > 0 d < z < 0 and z < d and filled with materials of permittivities ɛ m and ɛ respectively. The electric field produced by a dipole p placed at r = (0 0 z in a homogeneous medium of permittivity can be written as ] E homo = k p + (p ] e ik r r r r = i ˆ d k k π k p (p k ± ] k± e i(k R+k z z (S where R = xˆx + yŷ k = ω/c k = k k = (k k / k = k xˆx + k y ŷ k ± = k ± k ẑ the + and sign must be chosen for z > z and z < z respectively and the square roots are taken to yield a positive imaginary part. Noticing that the unit vector ˆk ± k± /k together with the s and p polarization vectors ê s = ( k yˆx + k x ŷ/k and ê ± p = (±k k k ẑ/(k k form a complete set of orthonormal vectors so that ˆk ± ˆk ± + ê s ê s + ê ± p ê± p is the 3 3 identity matrix we can rewrite eq S as ˆ E homo = ik d k (p ês ê s + (p ê ± p π k ] ê± p e i(k R+k z z. From this expression the magnetic field can be obtained via Ampere s law (i.e. H = E/(ik which by making the replacement ik ± inside the integral and using the identities ˆk ± ê s = ê ± p and ˆk ± ê± p = ê s Corresponding author: javier.garciadeabajo@nanophotonics.es

2 P (k dϕk P (k exp(ik R πj 0 k x iπk cos ϕ J k y iπk sin ϕ J cos ϕ (J 0 J + sin ϕ (J 0 + J ] k x πk k xk y πk sin(ϕ J cos ϕ (J 0 + J + sin ϕ (J 0 J ] k y πk TABLE S: Azimuthal integrals needed to evaluate eqs. S for different polynomials P (k. We use azimuthal coordinates k = (k ϕ k and R = (R ϕ as well as the abbreviation J m J m(k R. Adapted from Ref. ]. leads to H homo = ik π ˆ d k (p ês ê ± p k + (p ê± p ê ] s e i(k R+k z z. As we are interested in obtaining the reflected field produced by the dipole in region I we consider waves moving downward from the dipole position at r (i.e. ê vectors multiply them by the corresponding reflection coefficients r s and r p for s and p polarization and reverse their orientation toward the upward direction (i.e. ê +. Further taking the dipole position z 0 + immediately above the interface this procedure leads to the reflected fields E ref I ˆ = ik d k π H ref I = ik π rs (p ê s ê s + r p (p ê p k ] ê+ p e i(k R+k z (Sa ˆ d k rs (p ê s ê + p k + r p (p ê p ê ] s e i(k R+k z. (Sb We aim to study surface-polariton modes supported by the film in region II and specifically we concentrate on modes signaled by the poles of r p which include plasmons in thin films. Incidentally a similar study could be carried out for the poles of r s which describe for example some of the guided modes in sufficiently-thick or sufficiently-high-refractive-index planar waveguides. Consequently we dismiss r s terms in the present study and approximate r p R p k p /(k k p (eq 3 in the main text where k p is the in-plane wave vector of the mode under consideration. We first proceed by carrying out the integral over the azimuthal angle of k with the help of the expressions compiled in Table S. This integration generates Bessel functions J n (k R which have the asymptotic behavior J n (k R (πk R / e i(k R π/4 nπ/ + e i(k R π/4 nπ/ ] in the k R limit. For the remaining radial integral we proceed in a similar way as explained in the Methods section of the main text: the contribution to the field associated with the mode at k = k p is dominated by the pole in r p ; we then extend the integral down to k = and integrate in the complex k plane by closing contours in the upper and lower half-planes for the terms e ik R and e ik R respectively; finally noticing that Im{k p } > 0 only the first of these terms is found to make a contribution to the resulting scattered field which becomes I H scat I πe iπ/4 R p e kp(ir z/ω kp R πe 3iπ/4 e kp(ir z/ω R p kp R kk p kp 3 (ip + Ωp z ( + iωẑ (S3a Ω (ip + Ωp z where Ω = ik p / k k p fully captures the effect of retardation. Indeed one has Ω = in the quasistatic limit and in particular eq S3a then reduces to eq 4 of the main text. In what follows we adopt this limit but still retain an overall factor of k in the expressions for the magnetic field (eq S3b. We now proceed in a similar way to obtain the fields in regions II and III just be replacing the polarization vectors with those propagating in the corresponding media and by making use of the self-consistent transmission and reflection coefficients of the involved interfaces (see Section S. After some lengthy but straightforward ˆϕ (S3b

3 3 algebra the resulting scattered fields associated with the mode under consideration reduce to II = πe iπ/4 e ikpr { } ɛ ɛ m kp R k3 p (ip + p z B p + e kp(z+d ( + iẑ Bp e kpz ( iẑ H scat II = i πe iπ/4 ɛm III = πe iπ/4 T p e kp(ir+z+d ɛ ɛ kp R H scat III e ikpr kp R kk p B p + e kp(z+d + Bp e kpz]( ip + p z ˆϕ k3 p (ip + p z ( iẑ = i πe iπ/4 ɛ e kp(ir+z+d T p kp R kk p (ip + p z ˆϕ where the dimensionless coefficients T p and B ± p are implicitly defined as the residues in t p T p k p /(k k p and β ± p B ± p k p /(k k p which are good approximations to the coefficients t p and β ± p (eqs S4b and S5 in Section S near the mode pole k = k p. We are now prepared to calculate the time-averaged Poynting vector S scat = c/(π]re { (H scat }. Assuming that k p is approximately real the radial component is found to be S scat I = ωk4 p S scat II R R p e kpz( ip + p z = ωk4 p B p + e kp(z+d + Bp e kpz ( ip + p z R S scat III = ωk4 p R T p e kp(z+d( ip + p z. It should be noted that although the magnetic field is proportional to k and thus vanishes in the quasistatic limit the Poynting vector introduces a factor of c rendering a finite product kc = ω. Finally we obtain the power scattered by the dipole by integrating over the surface of a cylinder or large radius R centered at the dipole and oriented perpendicularly to the film: P scat = ˆ dz ˆ π 0 R dϕ S scat = πωkp 3 Λ ( p / + p z where Λ = R p + T p +e kpd k p d(b p + (Bp +Bp (B p + +sinh(k p d( B p + + Bp ]. When explicitly working out R p T p B p ± (eqs S6 and k p (eq in the main text this expression reduces to Λ = R p thus recovering eq 5 of the main text for the power scattered by the dipole into surface polaritons. S. FRESNEL COEFFICIENTS The calculations presented above involve the Fresnel coefficients of the film which one can calculate in a Fabry-Perot model from the reflection and transmission coefficients r νij and t νij for polarization ν =sp and incidence from each medium i on the interface with each of its surrounding media j. We consider homogeneous isotropic media i = m and in regions I II and III respectively where the central (film region has thickness d. The reflection and transmission coefficients of the film for incidence from the top medium then reduce to r ν = r νm + t νmr νm t νm e ikzmd (S4a r νm r νm eikzmd t ν = t νmt νm e ikzmd (S4b r νm r νm eikzmd where k zm = (k ɛ m k / is the out-of-plane light wave vector in medium m. In the central region m the field can be described as a superposition of waves propagating upward and downward with coefficients β ν + = t νmr νm e ikzmd r νm r νm eikzmd (S5a βν t νm =. r νm r νm eikzmd (S5b

4 Near a polariton mode of in-plane wave vector k p these coefficients exhibit a divergence that we isolate to obtain the fields scattered by a dipole (Section S. More precisely for the structure depicted in Figure b of the main text we find k p r p R p k k p 4 β ± p B ± p k p k k p where are the residues used in Section S. k p t p T p k k p R p = ( ɛ ɛ m k p d ɛ ɛ m T p = ɛ ɛ m ɛm ɛ k p d ( + ɛ m (ɛ m + ɛ Bp = ( ɛ ɛ m k p d + ɛ m B p + = ( ɛ ɛ m e kpd k p d ɛ m (S6a ] e kpd (S6b (S6c (S6d S3. RESULTS WITH RETARDATION The results presented in the main text are obtained within the quasistatic limit (c. A straightforward extension of the derivations presented in the Methods section shows that the main results of the paper can be rigorously amended to include retardation by just using a single correction factor Ω ik p k according to the the summary presented in Table S. Specifically the result σ max λ p /π for -particle surface-polariton scattering is maintained independent of the dielectric and geometrical properties of the film even when retardation is included. Likewise we also recover the results σ ext = α i σ max µ i σ scat = σ max ( αi µ i (a (b FIG. S: (a Retardation factor Ω = ik p/k as a function of the ratio k/k p = λ p/. (b Retardation factor for the lowest-energy plasmons sustained by gold (solid curves and silver (dashed curves films of different thicknesses d embedded in an ɛ = material. The inset shows the corresponding plasmon dispersion relations. The metals are described using a Drude-like model ɛ m = ɛ b ω bulk/ω(ω + iγ] with parameters ɛ b = 9.5 ω bulk = 9.06 ev and γ = 7 mev for gold; and ɛ b = 4.0 ω bulk = 9.7 ev and γ = mev for silver.

5 5 Point Scatterer Line Scatterer { max σ in coup σ scat σ ext Quasistatic Limit Including Retardation ( e k p(ir z ( e + iẑ k p(ir z/ω kpr + iωẑ kpr πe iπ/4 R ( p kp 3 ip ɛ + p z πe iπ/4 R p kp (ip 3 ɛ /Ω + p z ] ] R p (π 6 A θ α x ɛ 3/ λ 3 + A θ + α z R p (π 6 A θ α x p ɛ 3/ λ 3 p + Aθ + α z R p (π 7 αx + α z ] R p (π 7 αx / + α z ] ɛ λ 5 p R p 6π 3 Im{α x + α z} λ p µ x λ 3 p 4π 4 R p λ 3 p µ z 8π 4 R { p λ p π x z } λ 3 p π R p A 0 s-pol 4gɛ 3/ ( λp σ max σ in coup N polariton N photon σ in coup r { A p-pol θ = 0 { 3 A p-pol θ = 0 A 0 s-pol ɛ λ 5 p R p 6π 3 Im{α x/ω + α z} λ p λ 3 p 4π 4 R p λ 3 p 8π 4 R { p λ p π x z { λ 3 p A p-pol θ = 0 π R p A 0 s-pol ( { 3 4gɛ3/ λp A p-pol θ = 0 A 0 s-pol (ˆx + iẑekp(ix z (ˆx + iωẑekp(ix z/ω R p ɛ 3/ R p πkp (ip x + P z (π 5 A λ x + A z p i Rp (π 3 (A x + A z i Rp λ p µ x λ p (π 3 R p λ p µ z (π 3 R p { } max σ in coup ɛ λ p N polariton N photon g ( λp R p πkp (ip x/ω + P z R p (π 5 A ɛ 3/ λ x + ΩA z p (π 3 (A x/ω + A z λ p λ p (π 3 R p λ p (π 3 R p ɛ g λ p λ p λ 0 TABLE S: Summary of results in the main text (quasistatic limit and their generalization to include retardation (rigorous corrections in red where we have defined the parameter Ω ik p/k (= in the quasistatic limit with k = ω /c kp. We have also defined A θ A cos θ and A θ + A + sin θ which are used in σ in coup. indicating that Figure b of the main text is a universal result. A plot of Ω as a function of k/k p (Figure Sa shows that the quasistatic limit is an excellent approximation for k 0.k p (i.e. λ p 0.. This is for example the case in 4 nm silver and gold films above.5 ev plasmon energy (Figure Sb. S4. INFLUENCE OF SIZE AND SHAPE OF THE SCATTERER In the main text we consider -like scatterers. In this section we discuss the validity of this approximation when the scatterers are taken to be metallic particles of finite size. More precisely we study spheres and disks made of either gold or silver which we simulate numerically using the boundary element method (BEM 3]. A summary of the results is presented in Figure S. In particular Figure Sadgj shows the imaginary part of the effective polarizability normalized to the particle volume (solid curves as obtained by dividing the extinction cross-section by 8π /( ɛ. We now concentrate on the lowest-energy plasmon feature of each spectrum and analyze it by approximating the polarizability as α = p 0/ ω 0 ω iγ/ (similar to eq 9 in the main text where p 0 ω 0 and γ are fitting parameters. This leads to the Lorentzian profiles represented in Figure Sadgj as dashed curves. Given the relatively small size of the particles compared (S7

6 6 (a (b (c (d (e (f (g (h (i (j (k (l FIG. S: Size-dependence analysis of the polarizability of metallic spheres (a-f and 0-nm-high disks (g-l. (adgj Spectral variation of the polarizability (solid curves for particles made of (ag gold and (dj silver along with Lorentzian fits (eq S7 dashed curves of the lowest-energy resonance (ω = ω 0. (bh Variation of the resonance position ω 0 (left axis solid curves and width γ (right axis dashed curves extracted from the Lorentzian fits in (adgj. (ek Resonant value of the polarizability (Im{α(ω 0} as a function of particle diameter D. We normalize the polarizability either to the volume of the particles (left axis solid curves or to the maximum polarizability of a lossless two-level scatterer α max = 3λ 3 0/(6π 3 ɛ (right axis dashed curves. (ci Effective dipole moment p 0 extracted from the Lorentzian fits. (fl Dimensionless prefactor F defined in the main text (Methods assuming a polariton wavelength λ p = 00 nm. The metals are described using the same dielectric functions as in Figure S. with the resonance light wavelength retardation does not play a major role so plasmons show a nearly sizeindependent frequency and width although the latter exhibits a rapid increase for silver (gold spheres of diameter D above 30 nm (50 nm due to radiative losses. We find an optimum size for the spheres near the onset of radiative losses for which the peak polarizability reaches 5% of the maximum possible value for a dipolar scatterer (Figure Se see caption. Additionally for silver disks the peak polarizability reaches 90% of the maximum possible value (Figure Sk. We are ultimately interested in the parameter F = 8π 4 p 0/( γ in λ 3 p (see Methods which reaches a few 00s for silver disks of 00 nm diameter coupling to λ p = 00 nm plasmons. In Figure S3 we show BEM simulations for the extinction spectra offered by gold and silver oblate ellipsoids similar to those in the main text but now including the effect of retardation. By performing Lorentzian peak fits according to eq S7 we then obtain p 0 and from here F and 55 for gold and silver respectively assuming λ p = 00 nm in reasonable agreement with the values of 7 and 595 reported in the main text. A moderate plasmon redshift is observed due to retardation as well as an increase of broadening ( γ = 0.6 ev and 0. ev for gold and silver which we ignore in the estimate of F.

7 7 0 Au Ag 0 - Drude Exp. data Lorentzian Fit FIG. S3: Spectral dependence of the polarizability of gold (Au and silver (Ag oblate ellipsoids (00 nm diameter 0 nm height under illumination along their rotational axis. We present BEM numerical calculations using a Drude-like model (solid curves see parameters in Figure S and experimental data 4] (dashed curves for the metal dielectric function along with Lorentzian fits around the prominent dipole plasmon (dotted curves eq S7. We also show the corresponding values of the factor F γ film /γ in (see labels. S5. POLARIZABILITY OF A CIRCULAR HOLE We present in Figure S4 results for the polarizability associated with a circular hole perforated in a selfstanding homogeneous isotropic film of zero thickness in normalized units. The result is universal independent of material composition provided the hole is small compared with the light wavelength. The analytical approximation α x /max{ α x } π 3 a /λ p (dashed curve see main text is in excellent agreement with full numerical simulations (solid curve when the hole radius is small compared with the polariton wavelength Numerical Simulation Analytical Approximation FIG. S4: In-plane electrical polarizability of a circular hole perforated in a self-standing polariton-supporting zerothickness film as a function of hole radius a. The polarizability is normalized to the maximum possible value max{ α x } = λ 3 p/(4π 4 R p and the hole radius is normalized to the polariton wavelength λ p. We compare numerical simulations with the analytical approximation α x/max{ α x } π 3 a /λ p. ] García de Abajo F. J. Rev. Mod. Phys ] Blanco L. A.; García de Abajo F. J. Phys. Rev. B ] García de Abajo F. J.; Howie A. Phys. Rev. B ] Johnson P. B.; Christy R. W. Phys. Rev. B