Absorption by an optical dipole antenna in a structured environment

We compute generalized absorption and extinction cross-sections of an optical dipole nanoantenna in a structured environment. The expressions explicitly show the influence of radiation reaction and the local density of states on the intrinsic absorption properties of the antenna. Engineering the environment could allow to modify the overall absorption as well as the frequency and the linewidth of a resonant antenna. Conversely, a dipole antenna can be used to probe the photonic environment, in a similar way as a quantum emitter.


I. INTRODUCTION
It is well-known that the emission frequency and linewidth of a dipole quantum emitter is modified by its local environment [1][2][3][4]. The linewidth directly depends on the photonic Local Density Of States (LDOS) which accounts for the number of radiative and nonradiative channels available for the emitter to relax in the ground state. The change in the emission frequency and linewidth induce by the environment can be described by considering the transition dipole as a classical dipole oscillator [2,4]. Therefore similar behaviors are expected for a dipole antenna (or a nanoparticle) interacting with its environment, the involved dipole being in this case the induced dipole that is responsible for scattering and absorption. Indeed changes in the induced dipole dynamics in optical antennas have already been observed [5][6][7], and the parallel with spontaneous emission dynamics has been mentioned on a qualitative phenomenological ground. Energy shifts and linewidths of plasmonic nanoparticles have also been discussed recently, based on general properties of damped harmonic oscillators [8]. A connection between LDOS maps and the scattering pattern of plasmonic structures has been established in a specific imaging configuration [9]. In this context, it seems that a general discussion of the influence of the environment on the absorption of an optical dipole antenna or nanoparticle would be useful. The purpose of this paper is to address this question, using a rigorous framework based on scattering theory, and to illustrate the conclusions on a simple example.
In this work, we investigate the role of the environment on the absorption cross-section of an optical dipole antenna using a rigorous theoretical framework. We show that it is possible to modify the overall absorption and its spectral properties by engineering the environment, and in particular the photonic LDOS. Conversely, it is possible to probe the environment using a resonant nanoantenna, specific measurements being able to produce LDOS maps. In section II and III, we derive the exact expressions of the dressed electric polarizability of a dipole antenna in an arbitrary environment, and deduce the expression of the generalized absorption and extinction cross-sections. In section IV, we discuss qualitatively the physical mechanisms affecting both the resonance frequency and the linewidth of a resonant antenna, using a simplified model. In section V we study numerically a simple but realistic example, based on the rigorous expression derived in section II. This allows us to illustrate the general trends and to give orders of magnitude. In section VI we summarize the main conclusions. This section is devoted to the computation of the dressed polarizability of an optical dipole antenna or nanoparticle (i.e. the polarizability that accounts for the interaction with the environment). To proceed, we follow the same procedure that has been used previsouly to compute the polarizability in vacuum [10][11][12]. We consider an electrically small particle (the generic term particle will be used to denote either a subwavelength optical antenna or a nanoparticle) of volume V and of permittivity ǫ(ω) embedded in an arbitrary environment, the particle lying at position r 0 (see figure 1). We assume that position r 0 lies in vacuum (the local refractive index at r 0 is assumed to be unity). To describe light propagation in the environment, we use the electric dyadic Green function G which connects the electric field at position r to an electric dipole source at point r ′ through the relation E(r) = µ 0 ω 2 G(r, r ′ )p(r). We denote by E ext the field in the environment in the absence of the particle (exciting field). The total electric field E at point r and at frequency ω reads where k 0 = ω/c, c being the speed of light in vacuum. The approximation of electrically small particle amounts to considering that the electric field is uniform inside the particle. Under this condition, the expression of the total electric field inside the particle (at point r 0 ) becomes We now split the integral of the Green dyadic into its singular part −L/k 2 0 and its non-singular part V G reg (r 0 , r 0 , ω) where we have assumed that G reg (r 0 , r ′ , ω) is constant over the volume of the nanoparticle. Note that L is real since it corresponds to the singularity of the Green tensor at a point that lies in vacuum (the Green tensor is computed in the absence of the particle) [13][14][15]. Equation (2) becomes The expression of the polarizability follows be writing the induced electric dipole moment of the particle in the form The last line defines the dressed polarizability α(ω), that in the most general situation is a tensor. Inserting Eq. (3) into Eq. (4), one obtains A more useful expression of the dressed polarizability is obtained by defining a reference polarizablity α 0 (ω). A usual choice for this reference is the quasi-static polarizability of the particle in vacuum [12], that reads Note that in the case of a spherical particle, the singularity (or depolarization) dyadic is L = I/3 so that α 0 (ω) would simplify into the well-known (scalar) quasi-static expression α 0 = 3V (ǫ−1)/(ǫ+2). Using Eqs. (5) and (6), the dressed polarizability has the final form This expression is the main result of this section. It shows that the dressed polarizability of a particle depends on the environment, the influence of the environment being fully described by the non-singular part of the dyadic Green function V G reg (r 0 , r 0 , ω). Note that Eq. (7) is the explicit expression of the effective polarizability discussed in Ref. [4]. For a resonant antenna or nanoparticle (e.g., supporting a plasmon resonance), it is instructive to rewrite Eq. 7 in the form The resonance frequency of the dressed polarizability is solution of the equation The influence of the environment on the resonance lineshape is made explicit by this simple analysis.

III. GENERALIZED ABSORPTION AND EXTINCTION CROSS-SECTIONS
The expression of the dressed polarizability is the starting point to compute generalized absorption and extinction cross-sections in an arbitrary environment. To carry out this derivation, we start with the expression of the time-averaged power absorbed inside the particle, given by is the current density induced in the particle. As the electric field is assumed to be uniform inside the particle, the absorbed power becomes Using Eq. (4), it can be rewritten as To define a generalized absorption cross-section, we have to introduce an incident local energy flux φ ext . Since the exciting field at the position of the particle is E ext (r 0 , ω), we define the incident local energy flux using the expression for a plane wave φ ext = |E ext (r 0 , ω)| 2 /(2µ 0 c). This definition is arbitrary, but has the advantage to coincide with the standard one when the particle lies in a homogeneous medium. The generalized absorption crosssection is then given by the ratio P a /φ ext . Using the relation Im , we obtain the final expression of the generalized absorption cross-section Equation (12) is a central result of this paper. It deserves some remarks before we analyze its consequences. Although Eq. (12) involves tensor notations, the absorption cross-section σ a (ω) is a scalar quantity. Expression (12) is exact and has been obtained under the only assumption that the electric field is uniform inside the particle (approximation of electrically small particle). It can be used, together with Eq. (7), to discuss the influence of the environment on the optical properties of any particle (or antenna) satisfying this condition. For a nonabsorbing material, the imaginary part of the permittivity ǫ(ω) vanishes, and the quasi-static polarizability α 0 is real. The term Im[α 0 (ω)] in Eq. (12) implies σ a (ω) = 0, as it should be. Finally let us emphasize that the generalized absorption cross-section that we have defined really describes the change of the intrinsic absorption of the particle induced by the environment (or in other word of the absorption probability given a local incident power). Its proper normalization by the local incident energy flux clearly distinguishes this effect on the absorbed power from that due to a mere change of the local incident power. It is also important to stress that although the generalized absorption cross-section that we have defined is a scalar, it depends on the orientation of the local exciting electric field, that encodes the anisotropy of the environment. Following the same procedure, it is also possible to compute the extinction cross-section, starting from the expression of the power extracted from the external field by the nanoparticle. The latter can be written in the form [12]: One obtains As for the generalized absorption cross-section, this expression is exact under the assumption of an electrically small particle. In the following, we will focus our attention on the absorption cross-section, but the analyses and the general trends can be translated to the extinction situation, that can be relevant to specific experimental configurations and type of measurements.

IV. QUALITATIVE DISCUSSION
Equation (12) can be used to compute the absorption cross-sections in a given environment. For realistic geometries, the computation can only be performed numerically (we will study a simple example in section V). For example, it is possible to use an iteration scheme to solve numerically the Dyson equation which is the closed form of the equation governing the Green function similar to Eq. (1) [16]. Nevertheless, in order to get some insight based on simple analytical formulas, we will study an oversimplified situation. First, we consider a spherical nanoparticle of volume V , small enough to be consistent with the electric dipole approximation that we use throughout this work. For such a shape, the singular part of the Green tensor is simply given by L = I/3 [13,14]. Second, we assume that the nanosphere is embedded in an environment that preserves for G reg (r 0 , r 0 , ω) the same symmetry as that of free-space [i.e. G reg (r 0 , r 0 , ω) = G reg (r 0 , r 0 , ω)I]. This is an unrealistic hypothesis (it would be strictly valid only for a homogeneous medium or a medium with cubic symmetry) but we shall use it only to discuss qualitatively general trends. Under these hypotheses, the quasi-static polarizability reduces to α 0 (ω) = 3V [ǫ(ω) − 1]/[ǫ(ω) + 2], and the dressed polarizability takes the form The absorption cross-section becomes It is clear in this expression that in absence of polarization anisotropy induced by the environment, the absorption cross-section does not depend on the exciting field [this is not the case in Eq. (12)].
In order to get a simple model of a resonant optical dipole antenna, we consider a metallic nanoparticle described by a Drude permittivity ǫ(ω) = 1−ω 2 p /(ω 2 +iωγ) where ω p is the plasma frequency and γ the intrinsic collision rate that describes absorption losses in the bulk material. The real and imaginary parts of the Green tensor, that both influence the absorption cross-section, have a well-defined meaning. In order to makes this more explicit, we introduce the photonic LDOS ρ(r 0 , ω), connected to the imaginary part of the Green tensor by We also introduce φ(r 0 , ω) that describes the influence of the real part of the non-singular Green tensor in a similar way: Using these definitions, the dressed polarizability of the metallic nanoparticle reads where ω 0 = ω p / √ 3 is the plasmon resonance frequency of the bare nanoparticle in the quasi-static limit. This expression naturally leads to the introduction of an effective frequency Ω 2 eff (ω) = ω 2 0 {1 − πV ωφ(r 0 , ω)/2} such that the resonant frequency of the particle in the environment is solution of the equation Ω 2 eff (ω) − ω 2 = 0. Similarly, an effective linewidth γ eff (ω) = γ + πV ω 2 0 ρ(r 0 , ω)/2 can be introduced.
From Eqs. (12) and (19), we can obtain the expression of the absorption cross-section in the simplified scalar model: Equation (20), together with the expressions of Ω eff (ω) and γ eff (ω), shows that the real part of the Green function contributes to a change of the resonance frequency, while the imaginary part (the LDOS) changes the linewidth. This is the same behavior as that known for a dipole emitter (either quantum or classical), although in the present situation we deal with the dipole induced inside the particle by the external field. It is interesting to note that the effective linewidth γ eff (ω) can only be larger than the intrinsic linewidth γ because of its dependance on the LDOS which is a positive quantity. The resonant behavior of σ a (ω) and the influence of the LDOS [through γ eff (ω)] deserve to be analyzed more precisely. Due to the frequency dependence of both Ω eff (ω) and γ eff (ω), the resonance lineshape is in general not a Lorentzian profile. Moreover, Ω eff (ω) and γ eff (ω) are not independent, since the real and imaginary parts of the Green function are connected by Kramers-Kronig relations. It is nevertheless possible to derive the expression of the generalized absorption cross-section at resonance. The resonance frequency ω a satisfies dσ a (ω a )/dω = 0. As described in appendix B, using this implicit equation, it is possible to express σ a (ω a ) in the form where the superscript ′ denotes a first-order derivative. This expression shows that both the LDOS and its first-order derivative influence the amplitude of the generalized absorption cross-section, through γ 2 eff (ω a ) and γ ′2 eff (ω a ), respectively. An increase of both quantities tends to decrease the absorption cross-section. In the particular case of an environment for which the spectral dependence of the LDOS can be neglected [γ ′2 eff (ω) = 0], we end up with σ a (ω a ) ∝ 1/γ 2 eff (ω a ). This result can be qualitatively explained in simple terms. The first step in the absorption process by a metallic nanoparticle is the excitation of the conduction electron gas. Then relaxation can occur either by radiative (emission of scattered light) or non-radiative channels (absorption due to electron-phonon collisions). Increasing the photonic LDOS increases the weight of radiative channels and therefore reduces absorption. The role of the LDOS in this process is essentially the same as that in the spontaneous decay rate of a quantum emitter by coupling to radiation. In order to illustrate the effects discussed above on a real example, and to get orders of magnitudes (i.e. to establish the possibility of experiments), we study quantitaively in this section the generalized absorption crosssection of a silver nanosphere, with radius R = 15 nm, interacting with a flat perfectly conducting surface (perfect mirror). The geometry of the system is shown in Fig. 2. To describe the silver nanoparticle, we use the tabulated values of the bulk permittivity ǫ(ω) taken from Ref. [17].
To compute the generalized absorption cross-section of the nanosphere, we use the exact expression Eq. (12). In order to compute relative changes, we define the normalized cross-section σ n a (ω) = σ a (ω)/σ vac a (ω), where σ vac a (ω) is the absorption cross-section of the bare nanosphere in vacuum: The vacuum polarizability α vac (ω) is given by with α 0 (ω) = 3V [ǫ(ω) − 1]/[ǫ(ω) + 2] [11,12]. The calculation of σ n a (ω) requires the calculation of the exciting field E ext and of the Green function G reg (r 0 , r 0 , ω) in the geometry in Fig. 2. This is a straightforward application of the image method, given in appendix C for completeness.
We show in Fig. 3(a) the variations of normalized LDOS (ratio between the full LDOS and the LDOS in vacuum) versus both the wavelength and the distance d between the mirror and the center of the nanoparticle. Figure 3(b) (red solid line) displays a section corresponding to λ = 400 nm in Fig. 3(a). We observe the wellknown oscillations due to interferences between incident and reflected waves on the mirror [2]. In the near-field regime corresponding to d ≪ λ, the relative variations of the LDOS are on the order of 10 %. This plots of the LDOS will be helpful in the qualitative analysis of the variations of the generalized absorption cross-section.
To study the influence of the mirror on σ a (ω), we first consider an s-polarized illumination (the incident plane wave has an electric field linearly polarized along the direction y). In this case, the induced electric dipole in the nanosphere is oriented along y. As a consequence, σ a (ω) is independent on the direction of incidence (angle θ in Fig. 2). In Fig. 4(a), we represent the variations of the normalized absorption cross-section σ n a (ω) (generalized absorption cross-section divided by free-space crosssection) versus both the wavelength λ and the distance d between the mirror and the center of the nanoparticle. The plasmon resonance, corresponding to λ ≃ 360 nm, is visible for d < 100 nm. Figure 4(b) displays a section view of figure 4(a) at λ = 361 nm. We observe oscillations of the absorption cross-section, as a clear signature of the influence of the mirror. The relative variations are on the order of 5 % for distances d between 50 and 250 nm. The behavior of σ n a (ω) can be compared to that of the partial LDOS ρ yy (r 0 , ω) in Fig. 3(b). The oscillations are in opposition, in agreement with the qualitative analysis presented in section IV: An increase of the LDOS tends to decrease the absorption cross-section.
The case of an illumination with a p-polarized plane wave (i.e., with an electric field in the x − z plane) can be analyzed in a similar manner. In this case, the electric dipole induced in the nanosphere depends on the direction of incidence θ. We have chosen θ = 45 o for the sake of illustration (note that θ = 0 would lead to the same behavior as that observed with s-polarized illumination). Figures 5(a) and 5(b) are the same as Figs. 4(a) and 4(b), but for p-polarized illumination. We observe a similar behavior of σ n a , with oscillations corresponding to relative variations of a few percent. Close to resonance, and for d = 30 nm (which corresponds to strong nanoparticlemirror interaction in this simple system), the relative variation of the absorption cross-section is of the order of 20 %. Finally, let us note that significant changes are observed in the near-field regime only. For d > λ/2, the influence of interactions with the environment remains weak.

VI. CONCLUSION
We have described the influence of the environment on the absorption cross-section of an optical dipole antenna or a nanoparticle, based on a rigorous framework. We have derived a generalized form of the absorption cross-section, based on the only assumption that the electric field is uniform inside the antenna (electric dipole approximation). In the case of a resonant nanoparticle (plasmon resonance), we have analyzed qualitatively the role of the environment on the resonance frequency and on the linewidth. In particular, we have identified the role of the photonic LDOS, and shown that an increase of the LDOS results in a reduction of the generalized absorption cross-section. These effects have been illustrated on the simple example of a metallic nanoparticle interacting with a perfect mirror. In the field of optical nanoantennas, these results could be exploited along two directions. First, engineering the LDOS around an optical nanoantenna could allow some control of both the resonance frequency, and more interestingly on the level of absorption. Since high absorption remains a serious drawbacks of metallic nanoantenna, it might be possible to reduce absorption by an appropriate structuration of the environment. Second, measuring changes in the resonance lineshape of a metallic nanoparticle, as performed, e.g., in Ref. [6], should allow a direct mapping of the LDOS, without using fluorescent emitters. Finally, let us comment on the possibility of measuring σ a in situ. A potential method could be based on photothermal detection, in which a probe beam probes the temperature increase of the nanoparticle due to absorption. Such methods already offer the possibility of sensitive detection of nanoparticles in complex environments [19][20][21][22].
This work was supported by the EU Project Nanomagma under Contract No. NMP3-SL-2008-214107. E.C. acknowledges a doctoral grant from the French DGA.
Appendix A: Proof of the relation In this appendix, we give a proof of the relation The quasistatic polarizability is given by Eq. (6):  Fig. 2, the partial LDOS along x and y are equal. In all calculations, the minimum distance d ≃ 2R has been chosen at the limit of validity of the dipole approximation [18]. However, we also have Using Eqs. (A1) and (A2), we obtain the following relationship: which concludes the proof.
The resonance frequency ω a is defined by dσ a (ω a )/dω = 0. This gives us a relation satisfied by ω a . By factorizing by Ω 2 eff (ω a )−ω The computation of the modified absorption crosssection for a metallic nanosphere close to a perfect mirror requires the computation of the exciting field and of the Green tensor of the system composed by the mirror only. In the case of a s-polarised incident field given by E s inc (r, ω) = E 0 exp[ik x x + ik z z]e y where the incident wave-vector is k = k 0 (− sin θ, 0, − cos θ), the exciting field at the position r 0 of the nanosphere is simply given by the superposition of the incident and reflected fields: In the same way, in the case of a p-polarised incident field given by In order to express the Green function, we use the dipole image method. If the system consisting of the perfect mirror at z = 0 is illuminated by a source dipole p = (p x , p y , p z ) at position r 0 = (0, 0, z 0 ), the effect of the mirror can be replaced by the radiation of an image dipole p ′ = (−p x , −p y , p z ) placed at position r ′ 0 = (0, 0, −z 0 ). This allows us to compute the Green function of the system in a simple manner: G(r, r 0 , ω)p = G 0 (r, r 0 , ω)p + G 0 (r, r ′ 0 , ω)p ′ .
In this expression, G 0 is the Green tensor in vacuum given by with R = r − r 0 and u = R/R. PV denotes the principal value operator. At the position of the nanosphere (i.e. r = r 0 ), the singularity part of the Green tensor [needed to compute the quasi-static polarisability α 0 given by Eq. (6)] is simply L = I/3. The regular part [needed to compute the polarisability α given by Eq. (7)] is given by G reg (r 0 , r 0 , ω)p = G 0,reg (r 0 , r 0 , ω)p + G 0 (r 0 , r ′ 0 , ω)p ′ .