Fundamental Limits to the Coupling between Light and 2D Polaritons

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Fundamental Limits to the Coupling between Light and 2D Polaritons by Small Scatterers Eduardo J. C. Dias† and F. Javier García de Abajo*,†,‡ †

ICFO-Institut de Ciéncies Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, 08010 Barcelona, Spain

‡

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S Supporting Information *

ABSTRACT: Polaritonic modes in two-dimensional van der Waals materials display short in-plane wavelengths compared with light in free space. As interesting as this may look from both fundamental and applied viewpoints, such large confinement is accompanied by poor in/out optical coupling, which severely limits the application of polaritons in practical devices. Here, we quantify the coupling strength between light and 2D polaritons in both homogeneous and anisotropic films using accurate rigorous analytical methods. In particular, we obtain universal expressions for the cross sections associated with photon−polariton coupling by point and line defects, as well as with polariton extinction and scattering processes. Additionally, we find closed-form constraints that limit the maximum possible values of these cross sections. Specifically, the maximum photon-to-plasmon conversion efficiency in graphene is ∼10−6 and ∼10−4 for point and line scatterers sitting at its surface, respectively, when the plasmon and Fermi energies are comparable in magnitude. We further show that resonant particles placed at an optimum distance from the film can boost light-to-polariton coupling to order unity. Our results bear fundamental interest for the development of 2D polaritonics and the design of applications based on these excitations. KEYWORDS: 2D polaritons, graphene plasmonics, optical coupling, fundamental limits, optical theorem

T

unsuspected electronic and optical properties through nanostructured gating25 and layer stacking.26,27 Despite the benefits of achieving a strong spatial confinement, the small wavelength of 2D SPs (λp) compared to free-space light (λ0) implies a large in-plane momentum mismatch between them that needs to be compensated in order to enable the excitation of SPs through external illumination. A common approach to overcome this problem consists in using obstacles such as tips28 and patterned nanostructures29 that scatter light to produce induced evanescent fields with sufficient momentum to couple to propagating SP modes, effectively breaking the photon−polariton wavelength mismatch. In particular, point scatterers (e.g., molecules, defects, and nanoparticles) are typically employed as basic elements to mediate such coupling, and besides their potential for the design of practical applications, they further provide a simple reference to quantify light−polariton coupling. Importantly, as we show below, point scatterers additionally permit us to formulate fundamental limits to the efficiency of the coupling process.

he quest for optical excitations with increasing degree of spatial confinement has been recently fueled by the discovery of plasmons and other polaritonic modes sustained by atomically thin two-dimensional (2D) van der Waals materials, such as graphene1−7 and hexagonal boron nitride (hBN).8 These excitations display in-plane wavelengths in the range of a few nanometers at infrared frequencies, with a >100-fold reduction in mode wavelength relative to free-space radiation.9 As a result of such high confinement, 2D surface polaritons (SPs) encompass a large spatial concentration of electromagnetic energy that becomes appealing for producing intense optical nonlinearities10−12 and strong interaction with quantum emitters.13−16 SPs further display high sensitivity to the environment that becomes useful to optically detect and identify small amounts of organic17,18 and inorganic analytes.19 Additionally, because of the comparatively small number of atoms involved in a 2D nanostructure to support SPs, these excitations can be efficiently tuned by means of external stimuli, such as the potentials generated by electrical gates,2−4,20 the exposure to magnetic fields,21 and the introduction of optical heating,9,22,23 therefore holding great potential for applications in broad areas of optoelectronics.24 These means of control also enable the exploration of fundamental phenomena, which are further expanded by effectively creating materials that display © XXXX American Chemical Society

Received: December 7, 2018 Accepted: March 12, 2019

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DOI: 10.1021/acsnano.8b09283 ACS Nano XXXX, XXX, XXX−XXX

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Figure 1. Polaritons in 2D materials and scattering by point defects. (a) Schematic representation of the configurations under study. We consider a point scatterer (black dots) sitting on a thin film of finite thickness d. The scatterer is illuminated by either light or polaritonic waves (yellow), which produce an induced dipole that in turn gives rise to scattered polaritons (red). (b) Basic properties of a 2D polariton: wavelength λp, field-amplitude decay distance ∼λp/(2π) outside the film, and symmetry of in- and out-of-plane electric field components. (c) Dispersion relation of the plasmon modes supported by a thin metallic layer (solid curves) calculated in the quasistatic limit and compared with the wavelength obtained in the zero-thickness limit (dashed curves). The latter is obtained by condensing the film permittivity ϵm into a 2D conductivity (see main text). The upper horizontal axis gives the ratio ω/ωbulk corresponding to a Drude permittivity ϵm(ω) = 1 − ω2bulk/ω(ω + i/τ) for large relaxation time τ and ϵ1 = 1. Curves for dielectric contrast 1/2 and 1/10 are identical as those for 2 and 10, respectively. (d) Universal reflection-factor residue 9 p (eq 3) for finite thickness under the conditions of (c) (see main text). The plots in (c) and (d) only depend on the permittivity contrast ϵ1/ϵ2, for which several values are considered (see legend in (c)).

under the assumption that the SP modes are characterized by a large confinement ratio λ0/λp (see below and Supporting Information, SI). Then, the ratio λp/d of SP wavelength to film thickness is only a function of the permittivities of the materials inside and outside the film, as illustrated in Figure 1c for isotropic metallic films. But first, before considering films of finite thickness in more detail, we investigate the useful λp ≪ d limit. Polaritons in Atomically Thin Layers. Many of the properties of SPs in atomically thin layers (e.g., plasmons in graphene) can be accurately modeled in the zero-thickness limit, in which the material response is described via a frequencydependent 2D conductivity, σ(ω). In general, for a film material of permittivity ϵm and thickness d, we can implicitly define a 2D conductivity σ through the relation ϵm = 1 + 4πiσ/ωd. Figure 1c shows that this approximation (dashed curves) yields an accurate prediction of the SP wavelength down to λp ≳ 10d, a condition commonly fulfilled in few-atomic-layer films (d ∼ nm) displaying SPs with λp above a few tens of nanometers. We thus focus first on a film of zero thickness characterized by a 2D conductivity σ and placed in the z = 0 plane. The electric field associated with an SP propagating along the in-plane direction x then takes the form

In this article, we reveal important fundamental limits to the coupling of light to 2D SPs assisted by scatterers such as point and line defects. We present our results in the form of simple and rigorous analytical expressions, delivering an exact quantitative measure of photon-to/from-polariton coupling and polaritonto-polariton scattering. To that end, we first introduce a universal characterization of 2D SPs and their associated optical fields in terms of a single parameterthe ratio of their wavelength to the film thicknessregardless of the physical nature of the material’s response. This allows us to determine the efficiency of different optical scattering channels involving incident light and polaritons, expressed in terms of universal light−polariton coupling strengths and fundamental limits to the scatterer polarizability. In particular, we find the maximum possible photon-to-polariton cross section to be of the order of λ3p/λ0 and λ2p/λ0 for point and line scatterers, respectively. For graphene, the limits to in-coupling are quantified by the maximum possible ratio between the numbers of generated plasmons and incident photons Nplasmon/Nphoton ∼ α3 and α2 for point and line scatterers, where α ≈ 1/137 is the fine structure constant. Besides their fundamental interest, our results provide useful tools for the design of optical devices involving the in/outcoupling of propagating light and 2D SPs.

RESULTS AND DISCUSSION We first study the scattering of incident light or incident SPs by a point scatterer sitting at the surface of a thin film, as schematically sketched in Figure 1a. In particular, we consider the components of the scattered field that emerge from the scatterer as outgoing SPs. In what follows, we neglect retardation

Ep(r) = E0[x̂ + i sign{z}z]̂ ek p(ix−| z |)

(1)

which clearly satisfies the Poisson equation ∇·Ep = 0. Incidentally, we consider monochromatic fields of frequency ω with a temporal dependence given by E p (r, t) = 2Re{Ep(r)e−iωt}. B


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ACS Nano Table 1. Wavelength λp and Reflection Residue 9 p (eq 3) for SPs in Different Types of Films (See Methods)a

For the 2D film we define ϵ̅ = (ϵ1 + ϵ2)/2. For the anisotropic film we define ϵm = parts.

a

This electric field describes oscillations along the film with wave vector kp, corresponding to an SP wavelength

distance (for 1/e intensity decay) Lp = 1/(2 Im{kp}) = (ω − ωg)τλp/(4π) is large compared with λp. Polariton Coupling and Scattering. The above results can be readily extended to films of finite thickness d in the quasistatic limit (λp ≪ λ0), where d provides a natural length scale in the system, and indeed, we find that light coupling can be expressed in terms of universal functions of the dimensionless ratio λp/d, which is in turn determined by the permittivities in the film and the surrounding media, as illustrated in Figure 1c for a metallic film. We concentrate here on SPs with p polarization (e.g., plasmons), whose in-plane wave vector kp is identified from a pole in the corresponding Fresnel reflection coefficient,

{ }

λp = 2π /Re k p

while it also implies a decay in intensity by a factor 1/e at a distance λp/(4π) from the surface (Figure 1b). The parallel field component is continuous and symmetrically distributed on both sides of the film, whereas the perpendicular component is antisymmetric (Figure 1b) and presents a jump at the film plane z = 0 in order to fulfill the boundary condition Δ(ϵEz) = 4πρind, where the induced surface charge ρind is now determined from the induced surface current jind = σEx through the continuity equation ρind = (−i/ω)∂xjind; putting these expressions together, we readily derive the dispersion relation iϵ ω kp = ̅ 2πσ

rp ≈ 9 p

(2)

kp k − kp

(3)

where 9 p is a dimensionless residue that is equally determined by the dielectric response of the film and the surrounding materials (see Figure 1d). Specifically, in the zero-thickness limit, we have9 9 p = ϵ1 / ϵ ̅ , while closed-form analytical expressions are equally obtained for homogeneous isotropic and anisotropic films (see Table 1 and Methods). In order to estimate the ability of small scatterers to couple light into SPs, we consider a point scatterer placed right on top of the film in the upper medium of permittivity ϵ1 (see dot in Figure 1b). We describe the scatterer through an effective polarizability tensor α(ω), which permits writing the dipole induced in the scatterer in response to an external electric field Eext as p = α·Eext. The polarizability is effective because it is assumed to already include the self-consistent interaction with the film. The field scattered by this dipole contains components of momentum overlapping the SP dispersion relation, which originate from the pole in eq 3 (i.e., k∥ = kp), and, indeed, are the only ones surviving at distant in-plane regions. Specifically, for large in-plane distance R from the dipole (i.e., |kpR| ≫ 1) the field in medium 1 (z > 0) reduces to (see Methods)

where ϵ̅ = (ϵ1 + ϵ2)/2 incorporates the effect of the dielectric environment on either side of the film (Figure 1b) and we assume σ to be isotropic. We remark that the use of the quasistatic limit is well justified upon examination of the 2D conductivity of atomic layers, which is generally described by the expression σ=

ϵxϵz and take all square roots to yield positive imaginary

ωD ie 2 ℏ ω + i/τ − ωg

where the frequency ωD acts as a Drude weight (e.g., ωD = EF/ (πℏ) for graphene doped to a Fermi energy EF, and ωD = ℏnd/ m* for a Drude metal of thickness d, 3D carrier density n, and effective mass m*); the gap frequency ωg is associated with optical phonons or exitons but is zero in graphene and 2D metals; and τ is a phenomenological relaxation time. Indeed, upon insertion of this expression into the dispersion relation (eq 2), we find λp/λ0 = αX ≪ 1 for frequencies ω that make X = (2π/ ϵ ̅ )ωD/(ω − ωg) of order unity. In what follows, we further assume long relaxation time τ, so that the SP propagation C


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ACS Nano Escat = E Rscat(R̂ + iz)̂

ek p(iR − z) k pR

2 ϵ1 ωλp |E0|2 Ip = |9 p| (2π )3

(4)

We now express the induced dipole as p = E0(αxx̂ + iαzẑ) in eq 5, as determined by the incident plane wave (eq 1), leading to the scattering cross section

where E Rscat ̂ =

2π e−iπ /4

9p ϵ1

k p3(ip ·R̂ + pz )

scat σpoint [length]

and p∥ = (px, py). Equation 4 has a similar spatial dependence to that of eq 1, with x̂ replaced by the unit radial vector R̂ and further incorporating a 1/ R decay that guarantees energy conservation for 2D circular waves. We seek to quantify the strength of coupling from incident light to SP modes by calculating the cross section defined by the ratio of power Pscat carried by the scattered surface modes to the incident light intensity ϵ1 c |E ext|2 /(2π ). The former can be evaluated by computing the Poynting vector for a circularly scattered wave (eq 4), which requires going beyond the quasistatic limit. Instead, we use a computationally simpler, yet rigorous alternative argument based on the decay rate of the induced dipole, which leads to (see Methods) P scat =

|9 p| (2π )4 ω ϵ1

λp3

(|p |2 /2 + |pz |2 )

ext σpoint [length] =

(9)

|9 p| 16π 3 ϵ1

λp2

Im{αx + αz} (10)

which results from the surface-integrated total (i.e., incident +scattered) power flux along the forward-scattering direction (see Methods). Incidentally, the above results can be trivially generalized to include a finite distance z0 from the scatterer to the film by just multiplying the polarizability by e−2πz0/λp and also correcting the coefficients Aν=±, 0. The effect of film thickness and material composition is captured both in 9 p and in the frequency dependence of the λp/ d ratio. For a homogeneous film of nearly lossless metallic permittivity ϵm (i.e., Re{ϵm} < 0 and Im{ϵm} ≪ | ϵm|), the SPs are plasmon polaritons characterized by (see Methods and Table 1) ÄÅ ÉÑ ÅÅÅ (ϵ1 − ϵm)(ϵ2 − ϵm) ÑÑÑ ÑÑ λp/d = 4π /logÅÅÅ ÅÅÇ (ϵ1 + ϵm)(ϵ2 + ϵm) ÑÑÑÖ (11)

(5)

|9 p| (2π )6 3 ϵ13/2 λp λ 0

2 2 2 2 l o o o A −cos θ |αx| /2 + A+sin θ |αz| (p‐polarized light) ×m o o o A 0 |αx|2 /2 (s‐polarized light) n

|9 p|2 (2π )7 = (|αx|2 /2 + |αz|2 ) 2 5 ϵ1 λp

which has units of length and represents the portion of the incident SP that is scattered into directions other than the incident one. Additionally, we find it useful to obtain the extinction cross section for the incident SP:

and agrees with the computation based on the Poynting vector method (see Section S1 in the SI for a detailed calculation). This expression is dominated by surface polaritons arising from the pole of rp (see Methods). Now, expressing the dipole moment p in terms of the polarizability, we find that the point scatterer offers a photon-to-polariton (in-)coupling cross section in ‐ coup σpoint [area] =

(8)

and 9p =

where θ is the incidence angle relative to the film normal, we assume the polarizability to be diagonal in the xyz frame with inand out-of-plane components αx = αy and αz, respectively, and the correction factors (7a)

A 0 = |1 + rs|2

(7b)

(12)

which we plot in Figure 1c,d for selected values of the permittivity contrast ϵ2/ϵ1. Reassuringly, we recover the zerothickness limit (d → 0) by taking |ϵm| → ∞ while keeping ϵmd finite, which leads to λp/d → − πϵm/ ϵ ̅ from eq 11, and consequently 9 p → ϵ1/ ϵ ̅ (see Figure 1d). Table 1 also shows results for λp and 9 p corresponding to anisotropic films (see Methods), which directly apply to SPs in hyperbolic media.8 Optical Theorem and Fundamental Limits for Polariton Scattering. Powerful constraints are imposed on the polarizabilities of the scatterer from the fact that extinction is produced by both elastic scattering (i.e., a change in the propagation direction of the incident polariton) and inelastic absorption, so we must have σext ≥ σscat. Using eqs 9 and 10, this condition becomes Im{αx + αz}/(|αx|2 + 2|αz|2) ≥ 4π4 |9 p|/(ϵ1 λ3p), but in fact, separate conditions for in- and out-of-plane polarizability components can be extracted by considering two counter-propagating incident polariton waves with the same intensity, but with relative phases such that either the in- or the out-of-plane electric field component vanishes at the position of the scatterer. The induced dipole along the remaining nonvanishing component is increased by a factor of 2 relative to single-plane-wave incidence, leading to a 4-fold increase in the intensity of the elastically scattered component, also accompanied by a 2-fold increase in the extinction of each of the two

(6)

A± = |1 ± rp|2

λp ( −ϵmϵ1) πd ϵm2 − ϵ12

are introduced to account for the interaction with incident light reflected by the film, as obtained by using the Fresnel coefficients rs and rp for s and p polarization (see Methods). The cross section in eq 6 has units of area, so it can be interpreted as the portion of incident light plane wave that is effectively converted into SPs. We can proceed in a similar way to calculate the cross section σscat point offered by the scatterer toward an impinging SP wave (see eq 1) as the ratio Pscat/Ip, where Ip is the power flux carried by the SP plane wave (i.e., the power per unit of in-plane transversal length along y). We compute this flux by comparing eq 5 to the angular integral of the intensity associated with the scattered SP field (eq 4), which leads to (see Methods) D


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Figure 2. Fundamental limits to the polarizability and scattering cross sections. (a) The complex values of the polarizability are constrained to the shaded circle, whose diameter μi depends on the specific component: μz = ϵ1λ3p/(8π4 |9 p|) for αz and twice that value for αx. (b) Variation of the extinction, scattering, and absorption cross sections associated with the interaction of a polariton wave with a small scatterer as a function of the imaginary part of the scatterer polarizability. The horizontal axis is normalized to the maximum polarizability, whereas the vertical axis is normalized to the maximum extinction cross section.

incident waves. Putting these elements together, we find the conditions

depending on the orientation of the scatterer polarization. This result is universal, independent of thickness and dielectric properties of the film, and valid even when retardation is taken into consideration (see Section S3 in the SI). The difference between extinction and scattering represents absorption by the scatterer. For polarization only along one ext direction i = x or z, the absorption cross section σabs point = σpoint − σscat is thus the difference of two terms linear in Im{α } and |αi|2, point i respectively, so it admits a maximum as a function of the polarizability determined by ∂σabs point/∂αi = 0. This condition is trivially satisfied for Re{αi} = 0 and Im{αi} = μi/2, which leads to ext max{σabs point} = max{σpoint}/4. Interestingly, this value of the polarizability produces the same cross section for scattering and absorption, satisfying what is known as the critical-coupling condition. We find it interesting to present the extinction, scattering, and absorption cross sections normalized to the maximum extinction σmax ≡ max{σext point} as functions of the normalized imaginary part of the polarizability ζ = Im{αi}/μi, as shown in Figure 2b. Obviously, extinction corresponds to the straight line σext point/σmax = ζ. Also, for each value of ζ the minimum scattering and correspondingly the maximum absorption are obtained when 2 Re{αi} = 0, leading to the limiting values σscat point/σmax = ζ and 2 abs σpoint/σmax = ζ − ζ , respectively (see solid curves in Figure 2b). However, Re{αi} can take nonzero values, only limited by the condition that αi lies within the circle of Figure 2a, so by sweeping this parameter we obtain the colored regions presented in Figure 2b for the possible ranges of scattering and absorption cross sections. Incidentally, we have verified that Figure 2b remains valid even when including retardation (see Section S3 in the SI), so it can be regarded as a universal result, provided the extinction and scattering cross sections are proportional to Im{αi} and |αi|2, respectively, and the particle polarizability satisfies an opticaltheorem constraint of the form given by eq 13. This is, for example, the case of a particle in free space, for which the optical theorem is equally given by eq 13 with μi = 3λ30/(16π3).30 Limits to Light−Polariton Coupling. Inserting the maximum values of the polarizability obtained in the previous section (eq 14) into eq 6, we find the maximum normalincidence photon-to-polariton coupling cross section

l 4π 4|9 p| o o −1 | o Imo ≥ m } o o o ϵ1λp3 n αx o ~ l 8π 4|9 p| o o −1 | o Imo ≥ } m o o o ϵ1λp3 n αz o ~

where 9 p is defined in eq 3 and explicitly calculated for a metallic film in eq 12. These expressions, which constitute the optical theorem for 2D SPs, have the general form Im{−1/αi} ≥ 1/μi

(13)

with μi just differing by a factor of 2 between in- (i = x, μx = ϵ1λ3p/ (4π4|9 p |)) and out-of-plane (i = z, μz = ϵ1λ3p/(8π4|9 p|)) polarization components. Equation 13 can be recast as (Re{αi})2 + (Im{αi} − μi/2)2 ≤ (μi/2)2, which clearly reveals that the possible values of αi lie within a circle or radius μi/2 centered around αi = iμi/2 in the complex plane, as shown in Figure 2a. We conclude that the modulus of the polarizability and its imaginary part are both simultaneously maximized if Re{αi} = 0 and Im{αi} = μi, corresponding to the top point of the circle represented in that figure. Direct application of these results to a scatterer with polarization either parallel or perpendicular with respect to the film leads to the relation 3

max{|αz|} =

ϵ1λp 1 max{|αx|} = 2 8π 4|9 p|

(14)

The maximum achievable polarizabilty is thus larger for the inplane component (max{|αx|} > max{|αz|}), a result that we relate to its poorer coupling to SPs; indeed, a Lorentzian excitation in the scatterer decays more slowly to SPs for in-plane polarization, so it can have longer lifetime τ and, consequently, also larger onresonance polarizability ∝ τ. Additionally, for scatterers with the strongest possible polarizability αi = iμi, we find scat ext max{σpoint } = max{σpoint }

=

l o o 2, for x polarization ×m o o1, for z polarization π n

2λp

in ‐ coup }= max{σpoint

E

3 2 ϵ1 A 0 λp

π 2|9 p| λ 0

(15) DOI: 10.1021/acsnano.8b09283 ACS Nano XXXX, XXX, XXX−XXX

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Figure 3. Maximum photon-to-polariton coupling efficiency. (a) Schematic illustration of a tightly focused external light beam interacting with a point defect. The red circle represents the beam spot (area ∼ λ20), while the green circle is the coupling cross section (∼λ3p/λ0). (b) Maximum number of plasmons generated per incident photon Nplasmon/Nphoton for coupling to point (orange curves) or line (purple curves) defects in graphene as a function of photon energy normalized to the doping Fermi energy EF. We consider beams with either Gaussian (solid curves) or optimally focused (dashed curves) profiles.

results using the reciprocity theorem, or alternatively, by integrating the far-field Poynting vector generated by the scatterer upon SP irradiation (i.e., ϵ1 (2π )4 c |p|2 /(3λ 04) for upper-hemisphere emission). Dividing this result by the SP power flux Ip (eq 8), noticing that p = E0(αxx̂ + iαzẑ) is the dipole produced upon SP plane wave irradiation, and neglecting again surface reflection for simplicity, we find an associated cross section

This result imposes a severe reduction in the possible coupling of incident photons to polaritons when we compare it to the minimum focal spot in tightly focused light beams (see Figure 3a). We find it useful to estimate the maximum coupling efficiency by computing the ratio of polaritons produced per incident photon, Npolariton/Nphoton. This quantity must be equal to the ratio of SP-scattered to incident powers Pscat/Pbeam. However, the former is proportional to the light intensity at the position of the scatterer, which obviously depends on the angular profile of the light beam. We consider two relevant configurations for light focusing: a standard diffraction-limited Gaussian beam31 and an optimally focused beam that maximizes the intensity at the spot center, for which we find a ratio of focal-spot intensity to beam power given by |Ex|2 /P beam = gk 2 ϵ1 /c with g = 0.4646 and 2/3, respectively (see Methods), where we ignore reflection at the film surface for simplicity. Now, we evaluate Pscat using eq 5, where we consider an incident beam with electric field along x in the focal spot (i.e., |px | = A 0 |αxEx| and py = pz = 0). Using the maximum in-plane polarizability (eq 14), we find for the normalincidence coupling efficiency the fundamental limit Npolariton Nphoton

=

4gϵ13/2A 0 ijj λp yzz j z π |9 p| jjk λ 0 zz{

out ‐ coup σpoint [length] =

and from here, using eq 14, a maximum out-coupling efficiency out ‐ coup max{σpoint }=

Nphoton

= α3 graphene

4 4ϵ13/2 λp

3π 2|9 p| λ 03

is obtained for a scatterer with in-plane polarization and 1/4 times this value for out-of-plane orientation. Coupling through a Line Defect. Line defects offer a way of generating polariton plane waves and constitute interesting elements to control polariton propagation. For example, scattering by line defects in graphene has been considered following semianalytical methods.32,33 Here, we can obtain limits to polariton coupling and scattering by line defects by representing them as a chain of closely spaced dipoles, for which we can use the results obtained in the preceding paragraphs. We thus define the polarizability and dipole densities ( = α /L and 7⃗ = p/L for a line defect extending along y by normalizing to the defect length L. Assuming incident light or SP fields under normal incidence with respect to y for the sake of concreteness, the scattered field is easily obtained by integrating eq 4 along y in the |kpx | ≫ 1 limit, which produces an electric field like eq 1 with E0 replaced by

3

(16)

Because of the wide interest driven by plasmon polaritons in graphene, we now apply eq 16 to these excitations, whose wavelength depends on the doping Fermi energy EF as λp/λ0 = (2α/ ϵ ̅ )EF/ℏω.9 In this material, we can safely apply the zerothickness limit (9 p = ϵ1/ ϵ ̅ ) and neglect beam reflection (A0 = |1 + rs|2 = 1) to find Nplasmon

|9 p| (2π )6 (|αx|2 + |αz|2 ) 3 ϵ1 λp2λ 03

32g ϵ1 ij E F yz3 jj zz πϵ ̅ 2 k ℏω {

scat E line

which becomes on the order of α3 ∼ 10−6 for ℏω ∼ EF. This result is plotted in Figure 3b for both Gaussian and optimally focused beams, showing a dramatic boost in efficiency as the photon energy is reduced relative to EF, reaching a value Nplasmon/Nphoton ≈ 0.26% for self-standing graphene (ϵ1 = ϵ2 = 1) at ℏω = 0.1EF. The polariton-to-photon (out-)coupling efficiency is also an important magnitude that we can work out from the above

=

8π 3 9 p ϵ1λp2

(i7x + 7z) (17)

(see Methods). Obviously, polarization along y does not couple to SPs propagating along x. For SP incidence (see eq 1), the dipoles take the form 7x = (xE0 and 7z = i( zE0, which upon insertion into the scattered field permit us to obtain the reflection and transmission coefficients of the line scatterer, F


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ACS Nano scat 8π 3i9 p E line r= = (( x + ( z) E0 ϵ1λp2

Out-coupling from polaritons to photons is also limited by the optical theorem, leading to a maximum conversion fraction Nphoton/Npolariton = [ϵ1/(4 |9 p|)] (λp/λ0)2, as obtained by normalizing the upward power flux radiated by the line of induced dipoles (2π 3ω/λ 02)|7|2 (see Methods) to the SP plane wave power flux (eq 8), assuming the maximum possible polarizabilities |( i| = μi . We note that this fraction has the same order of magnitude and wavelength scaling ∼ (λp/λ0)2 as the incoupling fraction for a tightly focused light beam. Coupling through Linear Edges and Holes. The linear edge of a film constitutes a special case of line scatterer. In the zero-thickness limit for a self-sustained semi-infinite film, using the notation of eq 1, the polariton amplitude has been shown to satisfy the relation E0 = 2 E ext relative to a normally incident light field Eext.34 Converting this relation to a photon-topolariton cross section (with units of length, as it is normalized 2 to the edge length L) and using eq 8, we find σin‑coup edge = λp/(πλ0), in‑coup which coincides with max{σline } (with parameter ϵ1 = ϵ2 = 1 and 9 p = 1 appropriate for a self-standing thin film), indicating that the edge is already attaining the maximum possible coupling efficiency, with all of the generated polaritons obviously directed only along one direction away from the edge. A finite aperture in a film can also produce relatively efficient scattering. In particular, a circular hole perforated in a selfstanding zero-thickness film of conductivity σ effectively produces a negative current density j = −σE when exposed to an in-plane field E (i.e., compensating for the absence of a current in the hole area). For nonresonant holes of small radius a ≪ λp, we can approximate E as the externally incident field, leading to an induced dipole (i/ω)πa2 j, as obtained from the spatial integral of the current.30 Dividing by the field and using the relation between λp and σ given in Table 1, we find a polarizability αx ≈ a2λp/(4π) or, equivalently, αx/max{|αx|} ≈ π3(a/λp)2 (see eq 14), which reaches ∼30% of the maximum possible value when the radius a is 10% of the polariton wavelength λp, a result that we have confirmed though bruteforce numerical simulations (see Figure S4 in the Supporting Information). Optimization of Light−SP Coupling through Scatterer−Film Separation. An interesting observation comes from the fact that the point scattering efficiency depends on the separation z0 between the scatterer and the film surface through an exponential factor e−kpz0 ≈ e−2πz0/λp in the SP field. In particular, the scatterer dipole induced by an incident SP plane wave then becomes p = e−2πz0/λp E0(αxx̂ + iαzẑ). Repeating the analysis of the preceding sections with this factor in mind, we −4πz0/λp find the proportionalities σext Im{α} and σscat point ∝ e point ∝ −8πz0/λp 2 e |α| , and consequently, the optical theorem arising from 4πz0/λp scat , which produces σext point ≥ σpoint leads to max{α} ∝ e maximum possible extinction and scattering cross sections −4πz0/λp 2 independent of z0, while max{σin‑coup |α| ∝ e4πz0/λp point } ∝ e increases exponentially with the separation z0. Consequently, a finite z0 allows us to enhance the light−polariton coupling efficiency without affecting polariton−polariton processes. This result for the maximum possible in-coupling cross section is however conditioned to finding particles that can actually reach the predicted limit of max{α}. As we show below, an optimum distance can be found when considering a realistic model for the scatterer polarizability. In practice, actual scatterers (e.g., nanoparticles or molecules placed close to the film or even apertures, protrusions, and corrugations) present additional intrinsic loss channels, which limit the exponential

and t = 1 + r, respectively. We can now obtain limits to the line polarizability by imposing the condition of non-negative absorption 1 − |t|2 − |r|2 ≥ 0, which results in Im{(x + ( z}/(|(x + ( z|2 ) ≥ 8π 3|9 p| /(ϵ1λp2)

Arguing again that we can superimpose two counterpropagating SPs that cancel either the x or z field component, we obtain two individual conditions for the polarizabilities ( i expressed by eq 13 with max{|( i|} = μi = ϵ1λp2 /(8π 3|9 p|)

(18)

We note that this condition is the same for both i = x and i = z. Incidentally, a maximum possible absorption of 50% is achieved for r = −1/2, which implies Re{(x + ( z} = 0 and Im{(x + ( z} = μi /2 . This condition is compatible with the optical theorem just derived for the line scatterer (i.e., Im{( i} ≤ μi ). Additionally, we find |r| ≤ 1, with the equality being reached if the polarizability takes the maximum possible value ( i = iμi . The photon-to-polariton coupling cross section is now given as the ratio of the SP scattered power Pscat to the incident light intensity. Using eq 8 with E0 substituted by Escat line , we find P scat = 2 ×

Lϵ1ωλp2 8π 3|9 p|

scat 2 |E line |

where the leading factor of 2 accounts for the fact that the line scatters SPs toward either side of it. Considering normal incidence for simplicity, normalizing to the line length L, and scat ext noting that |Escat line | = |E | when Eline (eq 17) is evaluated for the maximum possible polarizability (eq 18), we find in ‐ coup }[length] = max{σline

2 ϵ1 λp

π |9 p| λ 0

for the maximum possible in-coupling cross section. We can also convert the above result into a photon-topolariton conversion efficiency by invoking the relation |Ex|2 L /P beam = gk /c

between the electric field intensity at the focus and the beam power per unit length L for Gaussian (g = 0.1887) and optimally focused (g = 1/4) line beams (see Methods) using the same procedure as for point scatterers. We find Npolariton/Nphoton = [(gϵ1/(π |9 p|)](λp/λ0)2. For graphene plasmons, this ratio becomes Nplasmon Nphoton

= α2 graphene

4g ij E F yz2 jj zz πϵ ̅ k ℏω {

which is on the order of α2 ∼ 10−4 for ℏω ∼ EF, but can be boosted at low frequencies as shown in Figure 3b. Comparing the value Nplasmon/Nphoton ≈ 0.16% obtained from this expression for ℏω = 0.1EF in self-standing graphene with 0.26% for point scatterers (see above), line scatterers are less efficient per photon at the energy under consideration and become even comparatively worse as the ratio ℏω/EF is reduced. G


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ACS Nano increase with z0 just predicted for σin‑coup point . In order to illustrate this point, we consider a particle hosting a spectrally isolated resonance. As shown in the Methods section, the particle polarizability including its interaction with the film can be approximated as α=

homogeneous ϵ1 dielectric material,36 where M is a local-field correction factor discussed in Methods), from which we find an increase in in-coupling cross section for in-plane polarization by 0 /(4γhomo) = [3π |9 p| /(16M 2 ϵ13/2)](λ 0 /λp)3. Multia factor γfilm plying this result by eq 15 and assuming M ≈ 1 and A0 ≈ 1 for 2 clarity, we obtain max{σin‑coup particle } ≈ 3λ0/(8πϵ1), which coincides with the maximum absorption cross section of a dipolar scatterer inside a medium of permittivity ϵ1 (incidentally this is in turn a factor of 4 smaller than the maximum possible extinction cross section of a lossless dipolar scatterer). We conclude that, in combination with tightly focused light beams, suitably chosen noble-metal and dielectric nanoparticles can be used to increase the photon−polariton in-coupling efficiency to unity order. Additionally, resonant molecules placed in a transparent host and held at cryogenic temperatures have been shown to perform as reasonably strong two-level systems,37 thus suggesting a possible realization of lossless scatterers for optimum light−SP in-coupling efficiency.

p02 /ℏ ω̃0 − ω − i(γin + γhomo + γfilm)/2

(19)

where p0 is an effective excitation dipole moment, ω̃ 0 is the resonance frequency after accounting for the shift produced by image interaction with the film, and the γ terms describe decay rates associated with different channels, namely, internal inelastic decay (γin), decay into radiation in the homogeneous ϵ1 host medium (γhomo), and coupling to the film dominated by SPs (γfilm). The latter admits the expression (see Methods) 0 − 4π z 0 / λ p γfilm = γfilm e

(20)

with 0 γfilm

=

|9 p| (2π )4 ϵ1 ℏλp3

2

CONCLUSIONS The analytical limits to the polarizability of point and line scatterers formulated above (see summary in Table 2) impose a

2

(|p0 | /2 + |p0z | )

which combined with eq 19 and neglecting γhomo as a radiative correction allows us to write the maximum polarizability (achieved at the resonance frequency ω = ω̃ 0) as

Table 2. Compilation of Wavelength Scalings for Several Relevant Coupling Magnitudes

3

ϵ1λp 1 1 max{|αz|} = max{|αx|} = e 4π z 0 / λ p 4 2 8π |9 p| (1 + γin /γfilm) (21)

magnitude = (C/|9 p|) × S

wavelength scaling S

point scatterer max{|αz|} max{σin‑coup point }

λ3p λ2p(λp/λ0)

2 ϵ1 A 0 /π 2

λp(λp/λ0) (λp/λ0)3

2 4ϵ3/2 1 /(3π ) 4gϵ3/2 /π 1

λ2p λp(λp/λ0)

ϵ1/(8π3) ϵ1 /π

(λp/λ0)2

gϵ1/π

max{σout‑coup point }

This result constitutes a generalization of eq 14, with which it coincides for nonlossy particles (γin = 0) and z0 = 0. Incidentally, in spite of the fact that our methods are based on classical electrodynamics, a factor 1/ℏ shows up in γfilm, which arises when following the semiclassical prescription of dividing the dipole-emitted power by the photon energy ℏω in order to convert it into a rate. The above analysis allows us to address the optimum choice of z0 needed to maximize the in-coupling cross section, which has 2 an explicit distance dependence given by σin‑coup point ∝ ξ|α| ∝ ξ (1 + −2 4πz0/λp 0 ξ γin/γfilm) , where ξ = e . This expression features a maximum as a function of ξ at ξ = γ0film/γin, or equivalently, z0 = [λp/(4π)] log(γ0film/γin), under the condition γ0film > γin, with an increase in σin‑coup by a factor ∼ γ0film/(4γin) relative to the point touching configuration (z0 = 0). As an example of a realistic scenario, we note that gold and silver disk-like nanoparticles of reduced size ( 0 and its reflection rp (−k∥kz1 + k2∥ẑ) × eik∥·R+ikz1z. In the quasistatic limit (c → ∞), we have ϵ1ω2 /c 2 − k 2 → ik

kz1 =

which permits us to write the sum of incident and reflected fields as −∇ϕ in terms of a potential ϕ = −k∥(ek∥z − rpe−k∥z)eik∥·R. We conclude that each component eik∥·(R−R′)+k∥(z−z′) in the incident potential of the dipole (eq 22) generates a reflected potential −rpeik∥·(R−R′)−k∥(z+z′). Performing this substitution in eq 22, we find the reflected potential in the z > 0 region ϕref

=

=

−1 (p·∇′) ϵ1

d2 k

∫ 2πk eik ·(R−R′)−k (z+z′)rp

1 (p ·∇R − pz ∂z) ϵ1

d2 k

∫ 2πk eik ·R−k (z+z )rp 0

(23)

where the second line is obtained by first making the replacement ∇′ → −∇R + ẑ∂z and then specifying the dipole position at r′ = (0, 0, z0). We now argue that SPs are signaled by a divergence of rp at k∥ = kp, so we use eq 3 to work out the k∥ integral analytically, leaving us with a I


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ACS Nano dimensionless factor 9 p that is a smooth function of k∥ and can be approximated by its value at k∥ = kp (see eq 3). Indeed, any k∥independent term in rp contributes as ∝1/|r − r′| to the integral in eq 23, or equivalently, an electric field that decays as ∼1/R3 at large inplane distances R; as we are interested in surface modes with slower decay ∼ 1/ R , such terms can be dismissed, and we further argue that any divergence-free term in rp will contribute negligibly at large distances. Integration over the angle of k∥ in eq 23 produces a Bessel function J0(k∥R) that in the |kpR| ≫ 1 limit leads to the asymptotic approximation47

d2 k

where Rdφ is the element of transversal length in the circular wavefront, and we replace the PW intensity |E0|2 by the scattered field intensity 2 |E Rscat ̂ | /|kpR| (cf. eqs 1 and 4). Inserting eq 4 in this expression, we find

ij 2π |9 p| yz Ip 5 zz k (|p |2 /2 + |pz |2 ) P scat = jjj j ϵ zz |E |2 p k 1 { 0 2

and comparison with eq 25 finally yields Ip = ϵ1ω|E0|2/(2π |9 p| k2p) and from here eq 8 with kp ≈ 2π/λp. Surface−Polariton Extinction Cross Section. We obtain the extinction cross section for an incident SP plane wave (eq 1) by integrating the total SP field intensity for large |kpR| near the forward scattering direction. In fact, we only need to consider the out-of-plane field component right outside the film, as the in-plane component has the same intensity and is just −π/2 out of phase, while the z dependence is common for incident and scattered waves. The integrated field intensity is then given by

∫ 2πk eik ·R−k (z+z )rp 0

≈ 9 pk p

∫0

∞

e−k (z + z 0) i(k R − π /4) [e + e−i(k R − π /4)] k − k p 2πk R dk

to which only nonresonant terms ∝1/r3 are incorporated by extending the range of integration down to −∞ and replacing k 2 + 0+ for k∥ in e−k∥z; then, by closing the contour in the upper (for the eik∥R term) or lower (for e−ik∥R) complex plane in k∥ and neglecting nonresonant contributions from branching points, the rp pole (in the upper plane because Im{kp} > 0) is found to contribute only through the outward wave as d2 k

∫ 2πk eik ·R−k (z+z )rp ≈ i9 pkp 0

∫

2 (|p |2 /2 + |pz |2 ) ϵ1ℏ

∫0

∞

2π k p(iR − z − z 0) − iπ /4 e k pR

k 2dk e−2k z 0 Im{rp}

2π |9 p| ϵ1

e−2k pz 0ωk p3 ( |p |2 /2 + |pz |2 )

∫

Noticing that the exponential term in this integral produces fast sign cancellations in the far-field limit (|kpR| ≫ 1) unless R ≈ x, we can approximate R − x ≈ y2/2x in the exponent, as well as R ≈ x and R̂ ≈ x̂ in the rest of the expression. Additionally, we set the dipole induced by the incident SP plane wave (see eq 1) as p = E0(αxx̂ + iαzẑ), which permits writing (see eq 4) ext σpoint =

(24)

l o −iπ /4 2 o Imo 9 pk p3(αx + αz) me o ϵ1 o o n

∫ dy

| o 2π ik py2 /2xo o e } o o k px o ~

With the change of variable θ = y k p/(2x) , we are left with the integral49 ∫ dθ exp(iθ 2) = π e iπ /4 , so the above expression directly leads to eq 10, where we approximate 9 p ≈ |9 p| and kp ≈ 2π/λp under the assumption of negligible film−material losses. Incidentally, for finite scattered-film separation z0, the above expression is simply corrected by a factor e−kpz0. Scattered SP and Light Fields Produced by a Line Scatterer. Each dipole element in a line scatterer extending along y contributes to the SP field −∇ϕref with a potential ϕref given by eq 23, in which 7⃗ dy′ (the dipole element along the line) must be substituted for p. The total field produced by the entire line of dipole elements is then obtained by integrating over y′, which produces δ(ky) and directly leads to

(25)

and from here, with kp ≈ 2π/λp and z0 = 0, we obtain eq 5. Power Flux Associated with an SP Plane Wave. We intend to find the power flux Ip carried by an SP plane wave (see eq 1), defined as the propagated power per unit of transversal length. Obviously, this quantity must be proportional to |E0|2; we can derive a general expression for it that applies to any type of film by considering the field scattered by a dipole (eq 4), approximating the circular SP wavefront as a plane wave for each direction R in the |kpR | ≫ 1 limit and integrating over in-plane scattering directions. More precisely, we have P scat = Ip

2

| l ik (R − x) o o o o E scat ̂ e p o ext σpoint = − 2Reo m dy R } o o o E k pR o o o 0 n ~

Approximating rp by the pole contribution of the dominant SP (plasmon-pole approximation, see eq 3) and noting that in the limit of negligible inelastic losses kp must have an infinitesimal positive imaginary part, we find Im{rp} ≈ π|9 p|kpδ(k∥ − kp), which upon insertion into eq 24 and multiplication by ℏω directly yields P scat =

e ik pR k pR

The square of the E0 term in this expression is independent of y (the inplane coordinate transversal to the propagation direction x), and therefore, it produces a contribution L|E0|2 proportional to the lateral beam size L. The square of the second term was considered above, integrated over scattering directions R̂ to yield the scattered power; however, near the forward direction, this term has a 1/R dependence that makes it vanish in the |kpR| ≫ 1 limit. The remaining interference term can be understood as the portion of incident beam that is extincted by interaction with the scatterer; when normalized to |E0|2, it should give a negative length that is subtracted from the L contribution of the direct beam, and consequently, it represents an extinction cross section with units of length, which we thus write as

Finally, using this expression in eq 23, performing the spatial derivatives via the substitution ∇R = ikpR̂ (i.e., neglecting terms that decay faster than 1/ R ), taking z0 = 0, and multiplying by −∇ = −ikp(R̂ + iẑ) to compute the scattered electric field from the potential, we readily obtain eq 4. We note that for finite dipole−film separation z0 the field carries an additional factor e−kpz0. Scattered SP Power Produced by a Point Dipole. Neglecting inelastic losses in the film, the power carried by the scattered SP field must be equal to Pscat = ℏωΓfilm, where Γfilm is the dipole decay rate. In a homogeneous dielectric, the decay rate (Γhomo ∝ 1/c3) is dominated by coupling to radiation and vanishes in the quasistatic limit under consideration, but in the vicinity of a film the reflected field provides a nonzero contribution arising from coupling to SPs. More precisely, we have36 Γfilm = (2/ℏ) Im{p*·Eref}, where Eref = −∇ϕref is evaluated at the position of the dipole r = r′ = (0, 0, z0). Using eq 23 for ϕref, we find48 Γfilm =

dy E0e ik px + E Rscat ̂

Escat =

1 (p·∇′)∇ ϵ1

dk

∫ |k x| eik (x−x′)−|k |(z+z′)rp x

x

x

(26)

In the spirit of the plasmon-pole approximation (eq 3), we write rp ≈ 29 p |kx|kp/(k2x − k2p), which is invariant under sign changes of kx and has the same pole structure as eq 3. Using this in eq 26 and integrating in the

∫ Rdφ |k1R| |ERscat̂ /E0|2 p

J


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ACS Nano

where β controls the degree of focusing. The field is maximized for β = 0, which produces a ratio |Ex |2 /P beam ≈ 0.4646k 2 ϵ1 /c . However, this is not the optimum solution that yields the maximum possible value of the ratio under consideration. Direct application of the Lagrange multiplier method yields the optimum choice ak∥s ∝ −kyk1/(k∥kz1) and ak∥p ∝ kx/k∥, which upon insertion in the above integrals yields

complex plane for kx, we find the scattered field to reduce to a plane scat = [8π 3 9 p/(ϵ1λp2)](i7x + 7z). wave like eq 1 with E0 replaced by E line A similar analysis can be carried out for the upward light emission from the line of dipoles, each of them contributing with a field proportional to k 2[7⃗ − (∇·7⃗ )∇]e ik1r /r , where k = ω/c and k1 = k ϵ1 . Integrating over y, this expression yields a field given by k 2[7⃗ − (∇· 7⃗ )∇]e ik1R + iπ /4 2π /(k1R ) , where R = x 2 + z 2 . Finally, integrating the Poynting vector resulting from this field over the upper hemisphere of emission directions, we find a radiated power (2π 3ω /λ 02)L |7|2 , which we use in the main text to calculate the polariton-to-photon coupling efficiency by a line scatterer. Polariton Dispersion Relation and Reflection Coefficients. A thin film of finite thickness with low-loss metallic permittivity (i.e., Re{ϵm} < 0 and Im{ϵm} ≪ −Re{ϵm}) supports plasmons, the dispersion relation of which is determined by the Fabry−Perot condition 1 − rp,m1rp,m2e2ikzmd = 0, where rp,mj = (ϵjkzm − ϵmkzj)/(ϵjkzm + ϵmkzj) is the Fresnel coefficient for reflection of p waves at the planar interface formed by the metal and the surrounding medium j, while

|Ex |2 /P beam = (2/3)k 2 ϵ1 /c . Incidentally, a similar procedure can be followed to include the effect of reflection by the film, simply by multiplying the coefficients ak∥σ in the expression for Ex by either 1 + rs or 1 − rp for σ = s or p polarization, respectively, which involves materials and thickness-dependent reflection coefficients rσ. For simplicity, we ignore this effect in our calculations of the |Ex|2/Pbeam ratio. We now repeat the above procedure for a line scatterer with translational invariance along y by considering only p waves with parallel wave vectors k∥ ∥ x̂. The ratio of the focal-spot intensity to the beam power now becomes |Ex|2L/Pbeam = 0.1887 k/c and |Ex|2L/Pbeam = k/4c for Gaussian and optimally focused beams, where L is the normalization scatterer length along y. Effective Polarizability of a Particle near a Film. The 3 × 3 polarizability tensor of a particle characterized by a resonance frequency ω0 can be approximated as50

kzj = k 2ϵj − k 2 is the normal light wave vector in medium j. We are interested in the λp ≪ λ0 limit, so we neglect retardation, which allows us to write rp,mj ≈ (ϵj − ϵm)/(ϵj + ϵm) and kzj ≈ ik∥, and from here the plasmon dispersion relation reduces to ÄÅ ÉÑ ÅÅÅ (ϵ1 − ϵm)(ϵ2 − ϵm) ÑÑÑ 1 Å ÑÑ kp = logÅ 2d ÅÅÅÇ (ϵ1 + ϵm)(ϵ2 + ϵm) ÑÑÑÖ

αpart =

t p,1mt p,m1rp,m2e2ikzmd 1 − rp,m1rp,m2e2ikzmd

(27)

where t p,mj = 2 ϵjϵm kz m/(ϵjkz m + ϵmkzj) ≈ 2 ϵjϵm /(ϵj + ϵm) and

t p, j m = (kzj/kz m)t p,mj ≈ 2 ϵjϵm /(ϵj + ϵm) are the in and out transmission coefficients at the jm interface. We then substitute rp,m1rp,m2 by e−2ikzmd evaluated at k∥ = kp (i.e., the noted Fabry−Perot condition for the SP mode) and expand everything to first order in k∥ − kp to produce the result shown in eq 12 and Table 1. For completeness, we note that the film reflection coefficient for s polarization is given by eq 27 with p replaced by s, rs,mj = (kzm − kzj)/ (kzm + kzj), ts,mj = 2kzm/(kzm + kzj), and ts,jm = 2kzj/(kzm + kzj), and in the quasistatic limit we have rs,mj ≈ 0 and ts,mj, ts,jm ≈ 1. For anisotropic films characterized by permittivities ϵx = ϵy and ϵz for in- and out-of-plane polarization, respectively, the above results need to be amended by (1) replacing ϵ m by ϵ x and (2) taking

where .·p is the additional surface-reflected component produced by the dipole and acting back on itself, which is obviously proportional to p through a 3 × 3 matrix .. The effective polarizability tensor that relates p to Einc then reduces to α=

sponds to the choice of coefficients

ak σ =

kz1/k1 e

−βk 2 / k 2

1 −.

−1 αpart

(29)

We readily obtain . from the surface reflection of the dipole field −∇ ϕref (see eq 23), which leads to ÄÅ É ÅÅ1/2 0 0 ÑÑÑ ÅÅ ÑÑ Å Ñ . = G ÅÅÅÅ 0 1/2 0 ÑÑÑÑ ÅÅÅ ÑÑ ÅÅÇ 0 0 1 ÑÑÑÖ with

k < k1

x̂ ·e1̂k+σ

(28)

p = αpart(Einc + .· p)

kz m = (k 2 − k 2/ϵz)ϵx for p polarization and kz m = k 2ϵx − k 2 for s polarization. An analysis similar to the homogeneous metallic film leads to the expressions for λp and 9 p given in Table 1. Maximum Field Produced by a Focused Light Beam. In order to quantify the maximum photon-to-polariton conversion efficiency, we first need to determine the ratio |Ex|2/Pbeam between the light intensity acting on the scatterer and the power Pbeam carried by the focused beam. We take the beam to be in a homogeneous medium of permittivity ϵ1 and consider a point scatterer at the focal point with in-plane polarization along x. The beam can be constructed as the superposition of plane waves of polarization σ (=s, p), parallel wave vector k∥, and unit polarization vector ê1+ k∥σ, with amplitudes ak∥σ such that the focal electric field reduces to E = ∑σ∫ k∥ 1 is the diameter-

The authors declare no competing financial interest.

ACKNOWLEDGMENTS F.J.G.A. would like to thank Dimitri Basov and Michael Fogler for stimulating and fruitful discussions. We thank Nader Engheta for suggesting the analytical hole model and Vahagn Mkhitaryan for help with numerical simulations of a hole. This work has been supported in part by the Spanish MINECO (MAT2017-88492R and SEV2015-0522), the ERC (Advanced Grant 789104eNANO), the Catalan CERCA Program, and Fundació Privada Cellex. E.J.C.D. acknowledges financial support through a “la Caixa” INPhINIT Fellowship Grant (ID 1000110434, No. LCF/BQ/DI17/11620057) and EU Marie Skłodowska-Curie grant (No. 713673).

to-height aspect ratio, and Δ = r 2 − 1 . We are now prepared to evaluate γ film /γ in with the help of eq 33, which leads to γfilm/γin = -(9 W/ϵ1)e−4πz 0/ λ p for p0 oriented parallel to the film (i.e., with the rotation axis of the prolate ellipsoid normal to the film and polarization along the diameter), where we have defined the prefactor

-=

8π 4p02 ℏγinλp3

=

π 3ϵ12(1 − ε)2 ω0 V (ϵb − ϵ1ε) γin λp3 L


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ACS Nano

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