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Nonlinear Mie theory for the second harmonic generation in metallic nanoshells Jérémy Butet, Isabelle Russier-Antoine, Christian Jonin, Noëlle Lascoux, Emmanuel Benichou, and Pierre-François Brevet* Laboratoire de Spectrométrie Ionique et Moléculaire, UMR CNRS 5579, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France *Corresponding author: [email protected]‑lyon1.fr Received April 20, 2012; revised June 27, 2012; accepted June 28, 2012; posted June 28, 2012 (Doc. ID 167217); published August 1, 2012 In this work, Mie theory extended to the specific case of the optical second harmonic generation (SHG) from metallic nanoshells is described. Our model results from a combination of the Mie theory developed for the linear optical response of concentric nanospheres and the Mie theory developed for the SHG from nanospheres. This approach leads to a multipolar expansion of the second harmonic scattered electric fields. The total scattered intensity and the relative contribution of each multipole to the scattered wave are directly calculated within this framework. Our model is then applied to the calculation of the second harmonic cross section for nanoshells made of the most common metals used in plasmonics, namely gold and silver. Finally, the effect of the aspect ratio, i.e., the ratio between the inner and the outer radii of the metallic nanoshell, a parameter that is known to greatly impact the surface plasmon resonance properties of the system, is discussed notably in terms of the tunability of the optical SHG from metallic nanoshells. © 2012 Optical Society of America OCIS codes: 190.2620, 190.4350, 240.4350, 290.4020.

1. INTRODUCTION Metallic nanoshells are nanoparticles composed of a dielectric inner core embedded in a thin metallic shell [1]. Their optical response is dominated by surface plasmon resonances corresponding to the collective excitations of the conduction band electrons in the metallic nanoshell. How the dimensions of the nanoshell affect the surface plasmon resonances properties is well understood [2–6]. A very useful approach for the description of the nanoshell optical response is to consider the hybridization model where the surface plasmon modes of a complex nanostructure are obtained by the combination of the surface plasmon resonances of its elementary components [2–4]. In this case, the surface plasmon resonances in metallic nanoshells arise from the interaction of the surface plasmon modes sustained by the outer and the inner shell surfaces and can be described as the hybridization between the surface plasmon modes of a metallic nanosphere and that of a nanocavity. The interaction strength depends on the ratio between the inner and the outer radii allowing for an efficient control of the optical response. From a practical point of view, plasmonic nanoshells are, as an example, interesting for plasmonic cancer therapy [7–13]. Over the last decade, an increasing amount of work has been devoted to the study of nonlinear optical processes in metallic nanostructures [14–16]. Experimentally, the most studied phenomenon is probably second harmonic generation (SHG), the nonlinear optical process whereby two photons at the fundamental frequency are converted into one photon at the second harmonic (SH) frequency. This phenomenon has been investigated in ensemble measurements, i.e., on a large number of nanoparticles simultaneously, as well as at the single particle level [17–27]. From a theoretical point of view, numerical simulations have been developed, especially in order 0740-3224/12/082213-09$15.00/0

to describe SHG in complex nanostructures [28–31]. However, even if theses simulations can yield a description of SHG from complex geometries, analytical or semi-analytical descriptions are also important and can lead to detailed complementary information on the physical processes involved. In particular, the analytical approach will provide a better understanding of the selection rules induced by symmetry properties or the exact origin of the nonlinear response for instance. For this reason, Mie theory has been extended to nonlinear optical processes like SHG, sum frequency generation or third harmonic generation [32–39] and to SHG from arrays of spheres [39,40]. Previously, Mie theory has also been developed to the case of the linear optical response from concentric nanospheres like nanoshells and core-shell nanostructures [41,42]. However, the nonlinear optical response, and the SHG one in particular, has never been performed in detail yet, even though the SH response from a metallic nanoshell has been discussed within the dipolar approximation [43]. Retardation effects are expected to play an important role for SHG since nanoshells are centrosymmetric nanostructures similar to nanospheres [32]. In this work, we therefore extend Mie theory to the case of SHG from metallic nanoshells taking into account retardation effects at both the excitation and emission steps, i.e., including higher modes than the dipolar one. Our model is based on the combination of Mie theory for the linear optical response of concentric nanospheres and that developed for the SHG response from nanospheres. The fundamental field is calculated first and then used to calculate the nonlinear polarization, i.e., the source to the SH fields. Applying the right set of boundary conditions, the scattered SH field is calculated allowing the determination of the SH scattering cross section. As an example, we apply our model to the © 2012 Optical Society of America

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calculation of the SH scattering cross section of silver and gold nanoshells.

Eshell ω

X X AE;shell ω l;m ∇ × j l k2 rXl;m k 2 l m1 AM;shell ωj l k2 rXl;m l;m

2. MODEL A. Calculation of the Fundamental Field In this section, we calculate the fundamental field inside the nanoshell. This problem has already been solved in the past using an expansion of the electromagnetic fields on the spherical harmonic basis vector set fXl;m g [41,42]. The incident electric field can be expressed as: Einc ω

X X AE;inc l;m ω ∇ × j l k3 rXl;m k3 l m1 AM;inc l;m ωj l k3 rXl;m ;

(1)

where k3 n3 ω ∕ c with n3 the refractive index of the surrounding media, and j l x are the spherical Bessel functions of the first kind. Without loss of generality, we consider that the electromagnetic field incident on the nanoshell is driven by a plane wave polarized along the x axis and propagating along the z axis (see Fig. 1). In this case of a plane wave polarized along the x axis and propagating along the z axis, the M;inc coefficients AE;inc are given by l;m and Al;m 1 l p AE;inc 4π2l 1δm;1 ; l;m ω i 2 m l p i 4π2l 1δm;1 : AM;inc l;m ω 2

(2)

AE;sca l;m ω AE;inc l;m ω

Hshell ω − Hcore ω × rc-s 0;

(8)

Esca ω Einc ω − Eshell ω × rs-sm 0;

(9)

(4)

Hsca ω Hinc ω − Hshell ω × rs-sm 0;

(10)

(5)

where rc-s rs-sm is the outward directed normal to the coreshell (shell-surrounding medium) interface. These boundary conditions allow the determination of the fundamental electric field everywhere inside the nanoshell [41,42]. The scattering coefficients are related to the one describing the incident wave and are expressed as [42]

∂ ∂r rj l k2 r

El ∂r∂ ryl k2 r − ∂r∂ rj l k3 rj l k2 r El yl k2 r AM;inc hl k3 ∂r∂ rj l k2 r El ∂r∂ ryl k2 r − ∂r∂ rhl k3 rj l k2 r E l yl k2 r l;m ω

AM;sca l;m ω

j l k3

where k1 n1 ω ∕ c and k2 n2 ω ∕ c. In Eqs. (4–6), n1 and n2 are the refractive indices of the nanocore and the nanoshell, respectively. yl x are the spherical Bessel functions of the second kind, and hl x are the spherical Hankel functions of the first kind. The expression of Esca ω corresponds to an outgoing spherical wave as required for a scattered wave. Also, Ecore ω is expressed with the spherical Bessel functions of the first kind j l x only because the spherical Bessel functions of the second kind yl x are not finite at the origin. This restriction does not apply to Eshell ω. The corresponding magnetic fields H are obtained using Faraday’s law as ∇ × E iωH in nonmagnetic media. The eight expansion coefficients obtained for each value of the couples of integers l; m are then calculated applying the boundary conditions at the core-shell and shell-surrounding media interfaces. Indeed, the tangential components of the electric and magnetic fields must be continuous at both interfaces:

Dl ∂r∂ ryl k2 r − ε2 ∂r∂ rj l k3 rj l k2 r Dl yl k2 r ; ε3 hl k3 r ∂r∂ rj l k2 r Dl ∂r∂ ryl k2 r − ε2 ∂r∂ rhl k3 rj l k2 r Dl yl k2 rrb ε3 j l k3 r

(6)

(3)

X X AE;core ω l;m ∇ × j l k1 rXl;m k1 l m1 AM;core ωj l k1 rXl;m ; l;m

BM;shell ωyl k2 rXl;m ; l;m

(7)

X X AE;sca l;m ω ∇ × hl k3 rXl;m Esca ω k3 l m1

Ecore ω

BE;shell ω l;m ∇ × yl k2 rXl;m k2

Eshell ω − Ecore ω × rc-s 0;

One can note that these expanding coefficients vanish for m ≠ 1 (see [42]). The scattered electric field, the electric fields inside the nanocore and in the nanoshell oscillating at the fundamental frequency are expressed as

AM;sca l;m ωhl k3 rXl;m ;

∂ ∂r rj l k2 r

rb

;

(11)

(12)

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from a bulk contribution due to field gradients [44,45]. In this present work, we only consider the pure surface contribution arising from both interfaces and therefore retain the χ ⊥⊥⊥ component. This element is indeed known to be the largest element of the surface susceptibility tensor in the case of metallic nanostructures [44,45]. Here, the symbol ⊥ denotes the direction normal to the interface. The introduction of the other tensor elements, however, is straightforward [33–34, 36] but not performed here for clarity of the analytical expressions. The two nonlinear polarization sheets present at both interfaces can be written as P c-s 2ω χ c-s E a ; ωE⊥ a ; ωδr − a− ; surf;⊥ ⊥⊥⊥ ⊥

(17)

P s-sm 2ω χ s-sm E b− ; ωE⊥ b− ; ωδr − b ; surf;⊥ ⊥⊥⊥ ⊥

(18)

Fig. 1. (a) Schema of the coordinate system. (b) Schematic representation of the nanoshell structure. The core, shell, and surrounding media are denoted by 1, 2, and 3, respectively.

with ε2 j l k2 ∂r∂ rj l k1 r − ε1 ∂r∂ rj l k2 r j l k1 r Dl ∂ ε ry k rj k r − ε y k r ∂ rj k r 1 ∂r

El

l

2

l

1

2 l

2

∂r

l

1

j l k1 r ∂r∂ rj l k2 r − ∂r∂ rj l k1 r j l k2 r ∂ rj k rj k r − j k r ∂ ry k r ∂r

l

1

l

2

l

1

∂r

l

2

;

(13)

ra

:

(14)

ra

As we will see below, the fundamental electric field components required for the nonlinear polarization calculation have to be evaluated inside the metallic shell. The coefficients necessary for the calculation of the SH sources are

where c-s and s-sm denote the core-shell and shell-surrounding medium interfaces, respectively. It is emphasized here that the fundamental fields are calculated inside the nanoshell, but the nonlinear polarizations are located outside the shell, as indicated by the superscripts and −. This is a standard description for the SHG response from metallic surfaces [46]. Both nonlinear polarization sheets, which are localized along the radial axis as seen with the Dirac delta function, can be expressed on the spherical harmonic basis set as P c-s 2ω surf;⊥

∞ X l X

C c-s l;m Y l;m θ; ϕ;

(19)

l0 m−l

− ε3 hl k3 k ∂r∂ ryl k2 r ε2 yl k2 r ∂r∂ rhl k3 r E;shell E;sca Al;m ω Al;m ω ∂ ∂ n3 n2 ∂r ryl k2 rj l k2 r yl k2 r ∂r rj l k2 r rb ε3 j l k3 r ∂r∂ ryl k2 r − ε2 yl k2 r ∂r∂ rj l k3 r ; n3 n2 ∂r∂ ryl k2 rj l k2 r yl k2 r ∂r∂ rj l k2 r

15

rb

ω BE;shell l;m

E;shell ∂ n2 ∂r∂ rj l k3 r − n2 ∂r∂ rhl k3 rAE;sca ω l;m ω − n3 ∂r rj l k2 rAl;m n3 ∂r∂ ryl k2 r

:

(16)

rb

B. Calculation of the Second Harmonic Sources The second step of our model is the determination of the nonlinear polarization oscillating at the SH frequency. As discussed in the introduction, SHG is forbidden in the bulk of centrosymmetric media within the dipolar approximation. Nevertheless, this symmetry is broken at the interface between two centrosymmetric media, thus allowing for SHG. Besides, it is known that the nonlinear polarization is also arising

2ω P s-sm surf;⊥

∞ X l X

C s-sm l;m Y l;m θ; ϕ;

(20)

l0 m−l

where the C l;m coefficients for both interfaces can be calculated using the spherical harmonics properties. After the re-incorporation of the expressions obtained for fundamental electric fields, these coefficients can be written as

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C c-s l;m χ ⊥⊥⊥

×

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X p l1 l1 1l2 l2 1 l1 ;m1 1 l2 ;m2 1

X X ASH;core 2ω l;m ∇ × j l K 1 rXl;m ; K1 l m0;2

Eshell 2ω

X X ASH;shell 2ω l;m ∇ × j l K 2 rXl;m K2 l m0;2

E;shell AE;shell l1 ;m j l1 k2 a Bl1 ;m yl1 k2 a k2 a

E;shell AE;shell l ;m j l2 k2 a Bl2 ;m yl2 k2 a × 2 k2 a Z × dΩY l;m Y l1 ;m1 Y l2 ;m2 ;

C s-sm l;m χ ⊥⊥⊥

×

(21)

X p l1 l1 1l2 l2 1 l1 ;m1 1 l2 ;m2 1

E;shell AE;shell l1 ;m j l1 k2 b Bl1 ;m yl1 k2 b k2 b

E;shell AE;shell l2 ;m j l2 k2 b Bl2 ;m yl2 k2 b k2 b Z × dΩY l;m Y l1 ;m1 Y l2 ;m2 :

×

(22)

dΩY l;m Y l1 ;m1 Y l2 ;m2

r 2l 12l1 12l2 1 4π l l1 l2 l l1 l2 × : m m1 m2 0 0 0

Eshell;∥ 2ω − Ecore;∥ 2ω −

l l1 l2

even;

m m1 m2 :

(25)

(26)

Since m1 and m2 can only take the 1 values (see above, Part A), only the modes for which m 0, 2 are required to write the two nonlinear polarizations. Having determined the expressions for the two SH polarization sheets, the SH electric field can be written everywhere in space. C. Calculation of the Second Harmonic Field The final step is the determination of the electric fields oscillating at the SH frequency in all space, namely within the nanoshell but also in the surrounding medium. The scattered electric field and the electric fields inside the nanocore and the nanoshell therefore can be written as Esca 2ω

X X ASH;sca 2ω l;m ∇ × hl K 3 rXl;m ; K3 l m0;2

(27)

(29)

(30)

(31)

at the core-shell interface and Esca;∥ 2ω − Eshell;∥ 2ω −

Hence, the following selection rules are immediately deduced from the Wigner 3j symbols’ properties: (24)

4π ∇ P c-s 2ω; ε1 2ω ∥ surf;⊥

Hshell;∥ 2ω − Hcore;∥ 2ω 0;

(23)

l ≤ l1 l2 ;

BSH;shell 2ω l;m ∇ × yl K 2 rXl;m ; K2

(28)

where K 1 n1 2ωk2ω, K 2 n2 2ωk2ω and K 3 n3 2ωk2ω with k2ω 2ω ∕ c. Similar to the case of the fundamental fields, the expansion coefficients for the harmonic electric fields are obtained applying the boundary conditions at both interfaces. The set of boundary conditions for a nonlinear polarization sheet standing at an interface has already been given by T. Heinz [47]. Considering a nonlinear polarization normal to the interface, it can easily be shown that only the modes with components of the electric fields normal to the interface can be excited. The boundary conditions for the SH fields are thus expressed as

All C l;m coefficients then vanish unless the integral R dΩY l;m Y l1 ;m1 Y l2 ;m2 is non-zero. This integral can be expressed as a product of Wigner 3j symbols [39]: Z

Ecore 2ω

4π ∇ P s-sm 2ω; ε3 2ω ∥ surf;⊥

Hsca;∥ 2ω − Hshell;∥ 2ω 0;

(32)

(33)

at the shell-surrounding medium interface, where ∥ denotes the component parallel to the interface. For each contributing mode, the boundary conditions lead to the following set of equations: ε1 2ω

j l K 1 rASH;core 2ω l;m K 1r

j l K 2 rASH;shell 2ω yl K 2 rBSH;shell 2ω l;m l;m ; (34) ε2 2ω K 2r ra

j l K 2 rASH;shell 2ω yl K 2 rBSH;shell 2ω l;m l;m K 2r hl K 3 rASH;sca 2ω l;m ; ε3 2ω K 3r rb

ε2 2ω

p 4πi ll1C c-s ASH;core 2ω ∂r∂ rj l K 1 r l;m − l;m K1 ε1 2ω

(35)

ASH;shell 2ω ∂r∂ rj l K 2 rBDH;shell 2ω ∂r∂ ryl K 2 r l;m l;m ; (36) K2 ra

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p 4πi ll1C s-sm ASH;sca 2ω ∂r∂ rj l K 3 r l;m l;m K3 ε3 2ω −

ASH;shell 2ω ∂r∂ rj l K 2 rBSH;shell 2ω ∂r∂ ryl K 2 r l;m l;m : (37) K2 rb

Since the spherical harmonics vectors are orthogonal, the excited modes are those for which the C l;m coefficients do not vanish. For this reason, the multipolar modes involved in the description of the SHG response from a core-shell nanostructure and from a homogeneous nanosphere are identical. The difference lies in the expansion coefficient expressions, i.e., the weight of each mode. Indeed, the nanoshell material properties only depend on the distance from the origin located at the center of the nanocore, not on the angles θ; φ. Both nanostructures possess the spherical symmetry. For each orthogonal mode, the four equations above allow the determination of the four unknown coefficients and their calculation is straightforward. As for the case of the linear electric fields, these coefficients give the SH electric field inside and outside the metallic nanoshell. Nevertheless, we will focus our attention on the SH scattered field which is the one experimentally observable. The corresponding coefficient is given by ASH;sca 2ω l;m

p T 4 C c-s 4πi ll 1 T 6 C s-sm l;m l;m − T 6 T 3 T 5 T 4 ε3 2ω ε1 2ω

(38)

with T1

ε2 2ωj l K 2 r ∂r∂ rj l K 1 r − ε1 2ωj l K 1 r ∂r∂ rj l K 2 r ; ε1 2ωK 2 2ωj l K 1 r ra (39)

T2

ε2 2ωyl K 2 r ∂r∂ rj l K 1 r − ε1 2ωj l K 1 r ∂r∂ ryl K 2 r ; ε1 2ωK 2 j l K 1 r ra (40)

ε2 2ωj l K 2 r ∂r∂ rj l K 3 r − ε3 2ωhl K 3 r ∂r∂ rj l K 2 r T3 ; ε2 2ωj l K 2 rK 3 rb (41) T4

j l K 2 r ∂r∂ ryl K 2 r − yl K 2 r ∂r∂ rj l K 2 r ; K2 rb

(42)

T5

n2 ε3 2ωhl K 3 b T ; n3 ε2 2ωj l K 2 b 1

(43)

T 6 −yl K 2 bT 1 j l K 2 bT 2 :

(44)

It is interesting to compare Eq. (38) with the expression obtained for the scattering coefficient in the case of SHG from a solid sphere. In order to perform this comparison, we consider a homogeneous sphere, i.e., a nanoshell with a shell and a core having identical properties. In this specific case, the interface

2217

between the core and the shell, and therefore the nonlinear polarization P c-s 2ω 0 associated, vanish. Since the shell surf;⊥ and core dielectric constants are similar, then T 1 T 5 0 and Eq. (38) reduces to 2ω ASH;sca l;m

p 4πi ll 1 C s-sm l;m : T3 ε3 2ω

(45)

As expected, this expression is identical to the one obtained in the case of the SHG from a solid homogeneous nanosphere [33,36]. This reduction shows the equivalence between our calculations and existing models. A rigorous validation of our model will be done in the near future by comparison with experimental measurements.

3. DISCUSSION In this part, the theory developed in the previous section is used to describe the SH response from silver and gold nanoshells beyond the dipolar approximation. We consider metallic nanoshells with an empty core, i.e., n1 1, dispersed in water, i.e., n3 1.33. A detailed analysis of the effect of the refractive index of the dielectric core and the embedding medium on the SHG from plasmonic nanoshells will be reported elsewhere. The dielectric constants for silver and gold are taken from [48]. The size effect is introduced for both silver and gold [5]. Indeed, the electron scattering at metal interfaces, in comparison to the corresponding metal bulk volume, increases for nanoparticles smaller than the electron mean free path, thereby modifying the surface plasmon lifetime and broadening the resonances [49]. In all our calculations, because of the thin nanoshells investigated, this effect is taken into account at both the fundamental and harmonic frequencies, using a surface scattering parameter A 1, a reduced electron mean free path d b − a introduced as the shell thickness and a Fermi velocity vf 1.4 106 m ∕ s taken with an identical value for gold and silver. The frequency dependence of the nonlinear susceptibility is also considered using a χ ⊥⊥⊥ component calculated using the following equation [44]: χ ⊥⊥⊥ −

aRS eε ε ω − 1 02 ; 4 r mω

(46)

where e and m denote the charge and the mass of the electron respectively, ε0 εr ω ε2 ω, and aRS is the so-called Rudnick-and-Stern parameter [50]. It is related to the harmonic nonlinear surface currents driven by the fundamental field in a direction normal to the interface. In a recent work, we have shown that in the case of silver nanoparticles this parameter is dispersion free as long as no interband transitions are involved at both the fundamental and harmonic frequencies [20]. The impact of these interband transitions on the Rudnick and Stern parameters is still unknown and the SHG intensity will be reported as relative values, the aRS parameter being fixed to unity in the present work. This hypothesis neither alters the relative weight of the dipolar and quadrupolar modes for a given fundamental wavelength nor their surface Plasmon resonance positions. The impact of these interband transitions on the Rudnick and Stern parameter can be readily introduced in the model. Finally, it is reminded that the interband transitions threshold is 3.85 eV (322 nm) in silver and 1.85 eV

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(670 nm) in gold. As in the case of the SHG from a metallic nanosphere, the total SH radiated power is an incoherent sum of contributions from the different contributing modes, even if interference between modes are revealed recording the SH wave in particular emission directions [18,51]: C SH sca 2ω

c

∞ X

8πK 23 2ω l;m

jASH;sca 2ωj2 : l;m

(47)

The contribution of the dipolar and quadrupolar modes to the total SH scattering cross section are calculated using l 1 and l 2, respectively. The octupolar (l 3) contribution to the total SH scattering cross section has been also calculated but has been found negligible for the nanoshell dimensions discussed in this work [18]. Note finally that we do not consider any breaking of the centrosymmetry through shape defects, for example, as could be the case in real nanoshells. Finally, the linear extinction cross section is also calculated for a direct comparison between linear and nonlinear optical properties. A. Silver Nanoshells The linear optical properties of silver nanoshells are discussed first. The calculations of the extinction cross section for a silver nanoshell (a 20 nm, b 30 nm) have been performed and two resonances are clearly visible. The resonance occurring at longer wavelength (close to 470 nm) corresponds to wavelength close to the dipolar surface plasmon resonance whereas the resonance occurring at shorter wavelength (close to 400 nm) corresponds to wavelength close to the quadrupolar plasmon resonance. The calculation of the SH scattering cross section for the same silver nanoshell (a 20 nm, b 30 nm) has been performed for a fundamental wavelength range from 350 nm up to 1200 nm. The results are depicted in Fig. 2(b). The SHG spectrum is dominated by four surface plasmon resonances corresponding to the enhancement of the dipolar emission mode only, the quadrupolar emission mode only, and the enhancement of both. The difference between both spectra is explained as follows. The most efficient mechanism leading to dipolar SH emission is the E 1 E 2 → E 1 mechanism where the two terms on the left of the arrow refer to the nature of the interaction with the fundamental wave and the third term describes the SH emission mode [33]. Hence, this notation describes here a dipolar emission (E 1 ) coming from the combination of an electric dipole (E1 ) and electric quadrupole (E 2 ) excitation. The pure dipolar mode E 1 E 1 → E 1 mechanism is forbidden for centrosymmetric objects, like the nanoshells discussed in this work, since Eq. (25) is not satisfied. SH dipolar emission needs retardation effects at the fundamental frequency, corresponding at the leading order to the quadrupolar mode, to be allowed. On the contrary, the quadrupolar SH emission mode can be excited without retardation effects at the fundamental frequency and arises from the E1 E 1 → E2 mechanism. Indeed, the retardation effects are included in the quadrupolar mode of the SH emission. The resonances labeled 3 and 4 in Fig. 2(b) correspond to wavelengths for which the harmonic frequency is resonant with the quadrupolar and with the dipolar bright mode, respectively. Only the mode with symmetry properties identical to the surface plasmon resonance is enhanced, e.g., the SH dipolar (quadrupolar) emission mode is enhanced

Fig. 2. (a) Extinction cross section as a function of the wavelength for a silver nanoshell (a 20 nm, b 30 nm). The dipolar and quadrupolar resonances are labeled 1 and 2, respectively; (b) Total SH scattering cross section calculated for the same silver nanoshell and for the dipolar (dashed line) and quadrupolar (dotted line) emission modes. Resonances are labeled from 1 to 4 (see main text for details).

when the harmonic frequency is tuned close to the dipolar (quadrupolar) surface plasmon resonance. On the other hand, both dipolar and quadrupolar emission modes are enhanced at resonances 1 and 2. In these cases, the fundamental wavelength is tuned close to the dipolar and quadrupolar resonances and both emission modes take advantage of the high fundamental electric field. It is interesting to note that the quadrupolar resonance labeled 1 in Fig. 2(b) arises from the E2 E 2 → E2 excitation channel as required by Eq. (25). As discussed in the introduction, the optical response of metallic nanoshells can be controlled by the ratio x between the inner radius a and the outer radius b. A calculation has thus been performed for different core sizes keeping the shell thickness constant at a value of 10 nm. Figure 3 shows the total SH scattering cross section for an inner radius equals to 10 nm, 20 nm, and 30 nm. As the core diameter increases, the coupling between the surfaces plasmons on the inner and the outer surfaces increases and all the resonances corresponding to symmetric coupling shift toward longer wavelengths, see arrows in Fig. 3. The position of the symmetric dipolar plasmon resonance shifts faster than the position of the quadrupolar one in this range of aspect ratio, x ranging from 0.41 up to 0.67. B. Gold Nanoshells In this part, the case of gold nanoshells is discussed in order to address similarities and differences with the silver ones. The

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Fig. 3. Total SH scattering cross section calculated for silver nanoshells with the same shell thickness (10 nm) but for increasing core dimensions. The evolution of resonance positions is shown by the arrows.

Fig. 5. Total SH scattering cross section calculated for gold nanoshells with the same shell thickness (10 nm) but for increasing core dimensions. The evolution of resonance positions is shown by the arrows.

calculations of the linear extinction cross section for a gold nanoshell (a 30 nm, b 40 nm) have been performed and only one resonance is observed [Fig. 4(a)]. Contrary to the case of silver nanoshells, the contribution of the quadrupolar mode to the whole extinction cross section is found negligible. Indeed, the amplitude of the quadrupolar surface plasmon resonance is far smaller than that of the dipolar surface plasmon

resonance. Figure 4(b) shows the total SH scattering cross section from the same gold nanoshell (a 30 nm b 40 nm) for a fundamental wavelength ranging between 500 nm and 1500 nm. Retardation effects are larger for short fundamental wavelengths and the SH scattering cross section increases as the fundamental wavelength decreases. As in the case of silver nanoshells, four resonances are observed and their physical origin, i.e., the excitation channels involved, is identical to the one discussed previously. In contrast, their position, width, and amplitude are different. In the case of gold nanoshells, the amplitude to width ratio is lower than in the case of silver nanoshells. This characteristic is mainly due to the increase of the surface plasmon damping induced by the creation of electron-hole pairs via interband transitions [52]. The nonlinear optical response of gold nanoshells is also tuned by changing the ratio x of the inner to outer core diameters as shown in Fig. 5. As in the case of silver nanoshells, the coupling between the surfaces plasmons on the inner and the outer surfaces increases as the inner radius a increases, leading to a redshift of the resonances corresponding to a symmetric coupling (see arrows in Fig. 5).

4. CONCLUSIONS

Fig. 4. (a) Extinction cross section as a function of the wavelength for a gold nanoshell (a 30 nm, b 40 nm). Only the dipolar resonance is visible because the amplitude of the quadrupolar one is too small. (b) Total SH scattering cross section calculated for the same gold nanoshell and for the dipolar (dashed line) and quadrupolar (dotted line) emission modes. Resonances are labeled from 1 to 4 (see main text for details).

In conclusion, the nonlinear Mie theory was extended to describe the case of SHG in metallic nanoshells, therefore including retardation effects at both the fundamental and harmonic wavelengths. Our model was applied to the calculation of total SH scattering cross sections showing that the contributions of the dipolar and the quadrupolar SH emission modes have a similar weight. The multipolar expansions of the electric field turn out to be very useful for the determination of both the allowed excitation channels and the mechanism of the scattered intensity enhancement. The impact of the aspect ratio was then discussed along with the wavelength dependence on the SH scattering cross section for gold and silver nanoshells. The effect of the dielectric constants of the core and of the surrounding medium on the SHG cross section from nanoshells can now be readily addressed with this model. This should allow further investigations, currently in progress, to discuss practical applications like sensing similarly to what has been proposed for homogeneous nanospheres [53].

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49. U. Kreibig and M. Vollmer, Optical Properties of Metal Cluster, Springer Series in Materials Science (Springer-Verlag, 1995. 50. J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971). 51. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975). 52. C. Sönnischen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002). 53. J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with multipolar second harmonic generation from spherical metallic nanoparticle,” Nano Lett. 12, 1697–1701 (2012).

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Nonlinear Mie theory for the second harmonic generation in metallic nanoshells Jérémy Butet, Isabelle Russier-Antoine, Christian Jonin, Noëlle Lascoux, Emmanuel Benichou, and Pierre-François Brevet* Laboratoire de Spectrométrie Ionique et Moléculaire, UMR CNRS 5579, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France *Corresponding author: [email protected]‑lyon1.fr Received April 20, 2012; revised June 27, 2012; accepted June 28, 2012; posted June 28, 2012 (Doc. ID 167217); published August 1, 2012 In this work, Mie theory extended to the specific case of the optical second harmonic generation (SHG) from metallic nanoshells is described. Our model results from a combination of the Mie theory developed for the linear optical response of concentric nanospheres and the Mie theory developed for the SHG from nanospheres. This approach leads to a multipolar expansion of the second harmonic scattered electric fields. The total scattered intensity and the relative contribution of each multipole to the scattered wave are directly calculated within this framework. Our model is then applied to the calculation of the second harmonic cross section for nanoshells made of the most common metals used in plasmonics, namely gold and silver. Finally, the effect of the aspect ratio, i.e., the ratio between the inner and the outer radii of the metallic nanoshell, a parameter that is known to greatly impact the surface plasmon resonance properties of the system, is discussed notably in terms of the tunability of the optical SHG from metallic nanoshells. © 2012 Optical Society of America OCIS codes: 190.2620, 190.4350, 240.4350, 290.4020.

1. INTRODUCTION Metallic nanoshells are nanoparticles composed of a dielectric inner core embedded in a thin metallic shell [1]. Their optical response is dominated by surface plasmon resonances corresponding to the collective excitations of the conduction band electrons in the metallic nanoshell. How the dimensions of the nanoshell affect the surface plasmon resonances properties is well understood [2–6]. A very useful approach for the description of the nanoshell optical response is to consider the hybridization model where the surface plasmon modes of a complex nanostructure are obtained by the combination of the surface plasmon resonances of its elementary components [2–4]. In this case, the surface plasmon resonances in metallic nanoshells arise from the interaction of the surface plasmon modes sustained by the outer and the inner shell surfaces and can be described as the hybridization between the surface plasmon modes of a metallic nanosphere and that of a nanocavity. The interaction strength depends on the ratio between the inner and the outer radii allowing for an efficient control of the optical response. From a practical point of view, plasmonic nanoshells are, as an example, interesting for plasmonic cancer therapy [7–13]. Over the last decade, an increasing amount of work has been devoted to the study of nonlinear optical processes in metallic nanostructures [14–16]. Experimentally, the most studied phenomenon is probably second harmonic generation (SHG), the nonlinear optical process whereby two photons at the fundamental frequency are converted into one photon at the second harmonic (SH) frequency. This phenomenon has been investigated in ensemble measurements, i.e., on a large number of nanoparticles simultaneously, as well as at the single particle level [17–27]. From a theoretical point of view, numerical simulations have been developed, especially in order 0740-3224/12/082213-09$15.00/0

to describe SHG in complex nanostructures [28–31]. However, even if theses simulations can yield a description of SHG from complex geometries, analytical or semi-analytical descriptions are also important and can lead to detailed complementary information on the physical processes involved. In particular, the analytical approach will provide a better understanding of the selection rules induced by symmetry properties or the exact origin of the nonlinear response for instance. For this reason, Mie theory has been extended to nonlinear optical processes like SHG, sum frequency generation or third harmonic generation [32–39] and to SHG from arrays of spheres [39,40]. Previously, Mie theory has also been developed to the case of the linear optical response from concentric nanospheres like nanoshells and core-shell nanostructures [41,42]. However, the nonlinear optical response, and the SHG one in particular, has never been performed in detail yet, even though the SH response from a metallic nanoshell has been discussed within the dipolar approximation [43]. Retardation effects are expected to play an important role for SHG since nanoshells are centrosymmetric nanostructures similar to nanospheres [32]. In this work, we therefore extend Mie theory to the case of SHG from metallic nanoshells taking into account retardation effects at both the excitation and emission steps, i.e., including higher modes than the dipolar one. Our model is based on the combination of Mie theory for the linear optical response of concentric nanospheres and that developed for the SHG response from nanospheres. The fundamental field is calculated first and then used to calculate the nonlinear polarization, i.e., the source to the SH fields. Applying the right set of boundary conditions, the scattered SH field is calculated allowing the determination of the SH scattering cross section. As an example, we apply our model to the © 2012 Optical Society of America

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calculation of the SH scattering cross section of silver and gold nanoshells.

Eshell ω

X X AE;shell ω l;m ∇ × j l k2 rXl;m k 2 l m1 AM;shell ωj l k2 rXl;m l;m

2. MODEL A. Calculation of the Fundamental Field In this section, we calculate the fundamental field inside the nanoshell. This problem has already been solved in the past using an expansion of the electromagnetic fields on the spherical harmonic basis vector set fXl;m g [41,42]. The incident electric field can be expressed as: Einc ω

X X AE;inc l;m ω ∇ × j l k3 rXl;m k3 l m1 AM;inc l;m ωj l k3 rXl;m ;

(1)

where k3 n3 ω ∕ c with n3 the refractive index of the surrounding media, and j l x are the spherical Bessel functions of the first kind. Without loss of generality, we consider that the electromagnetic field incident on the nanoshell is driven by a plane wave polarized along the x axis and propagating along the z axis (see Fig. 1). In this case of a plane wave polarized along the x axis and propagating along the z axis, the M;inc coefficients AE;inc are given by l;m and Al;m 1 l p AE;inc 4π2l 1δm;1 ; l;m ω i 2 m l p i 4π2l 1δm;1 : AM;inc l;m ω 2

(2)

AE;sca l;m ω AE;inc l;m ω

Hshell ω − Hcore ω × rc-s 0;

(8)

Esca ω Einc ω − Eshell ω × rs-sm 0;

(9)

(4)

Hsca ω Hinc ω − Hshell ω × rs-sm 0;

(10)

(5)

where rc-s rs-sm is the outward directed normal to the coreshell (shell-surrounding medium) interface. These boundary conditions allow the determination of the fundamental electric field everywhere inside the nanoshell [41,42]. The scattering coefficients are related to the one describing the incident wave and are expressed as [42]

∂ ∂r rj l k2 r

El ∂r∂ ryl k2 r − ∂r∂ rj l k3 rj l k2 r El yl k2 r AM;inc hl k3 ∂r∂ rj l k2 r El ∂r∂ ryl k2 r − ∂r∂ rhl k3 rj l k2 r E l yl k2 r l;m ω

AM;sca l;m ω

j l k3

where k1 n1 ω ∕ c and k2 n2 ω ∕ c. In Eqs. (4–6), n1 and n2 are the refractive indices of the nanocore and the nanoshell, respectively. yl x are the spherical Bessel functions of the second kind, and hl x are the spherical Hankel functions of the first kind. The expression of Esca ω corresponds to an outgoing spherical wave as required for a scattered wave. Also, Ecore ω is expressed with the spherical Bessel functions of the first kind j l x only because the spherical Bessel functions of the second kind yl x are not finite at the origin. This restriction does not apply to Eshell ω. The corresponding magnetic fields H are obtained using Faraday’s law as ∇ × E iωH in nonmagnetic media. The eight expansion coefficients obtained for each value of the couples of integers l; m are then calculated applying the boundary conditions at the core-shell and shell-surrounding media interfaces. Indeed, the tangential components of the electric and magnetic fields must be continuous at both interfaces:

Dl ∂r∂ ryl k2 r − ε2 ∂r∂ rj l k3 rj l k2 r Dl yl k2 r ; ε3 hl k3 r ∂r∂ rj l k2 r Dl ∂r∂ ryl k2 r − ε2 ∂r∂ rhl k3 rj l k2 r Dl yl k2 rrb ε3 j l k3 r

(6)

(3)

X X AE;core ω l;m ∇ × j l k1 rXl;m k1 l m1 AM;core ωj l k1 rXl;m ; l;m

BM;shell ωyl k2 rXl;m ; l;m

(7)

X X AE;sca l;m ω ∇ × hl k3 rXl;m Esca ω k3 l m1

Ecore ω

BE;shell ω l;m ∇ × yl k2 rXl;m k2

Eshell ω − Ecore ω × rc-s 0;

One can note that these expanding coefficients vanish for m ≠ 1 (see [42]). The scattered electric field, the electric fields inside the nanocore and in the nanoshell oscillating at the fundamental frequency are expressed as

AM;sca l;m ωhl k3 rXl;m ;

∂ ∂r rj l k2 r

rb

;

(11)

(12)

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from a bulk contribution due to field gradients [44,45]. In this present work, we only consider the pure surface contribution arising from both interfaces and therefore retain the χ ⊥⊥⊥ component. This element is indeed known to be the largest element of the surface susceptibility tensor in the case of metallic nanostructures [44,45]. Here, the symbol ⊥ denotes the direction normal to the interface. The introduction of the other tensor elements, however, is straightforward [33–34, 36] but not performed here for clarity of the analytical expressions. The two nonlinear polarization sheets present at both interfaces can be written as P c-s 2ω χ c-s E a ; ωE⊥ a ; ωδr − a− ; surf;⊥ ⊥⊥⊥ ⊥

(17)

P s-sm 2ω χ s-sm E b− ; ωE⊥ b− ; ωδr − b ; surf;⊥ ⊥⊥⊥ ⊥

(18)

Fig. 1. (a) Schema of the coordinate system. (b) Schematic representation of the nanoshell structure. The core, shell, and surrounding media are denoted by 1, 2, and 3, respectively.

with ε2 j l k2 ∂r∂ rj l k1 r − ε1 ∂r∂ rj l k2 r j l k1 r Dl ∂ ε ry k rj k r − ε y k r ∂ rj k r 1 ∂r

El

l

2

l

1

2 l

2

∂r

l

1

j l k1 r ∂r∂ rj l k2 r − ∂r∂ rj l k1 r j l k2 r ∂ rj k rj k r − j k r ∂ ry k r ∂r

l

1

l

2

l

1

∂r

l

2

;

(13)

ra

:

(14)

ra

As we will see below, the fundamental electric field components required for the nonlinear polarization calculation have to be evaluated inside the metallic shell. The coefficients necessary for the calculation of the SH sources are

where c-s and s-sm denote the core-shell and shell-surrounding medium interfaces, respectively. It is emphasized here that the fundamental fields are calculated inside the nanoshell, but the nonlinear polarizations are located outside the shell, as indicated by the superscripts and −. This is a standard description for the SHG response from metallic surfaces [46]. Both nonlinear polarization sheets, which are localized along the radial axis as seen with the Dirac delta function, can be expressed on the spherical harmonic basis set as P c-s 2ω surf;⊥

∞ X l X

C c-s l;m Y l;m θ; ϕ;

(19)

l0 m−l

− ε3 hl k3 k ∂r∂ ryl k2 r ε2 yl k2 r ∂r∂ rhl k3 r E;shell E;sca Al;m ω Al;m ω ∂ ∂ n3 n2 ∂r ryl k2 rj l k2 r yl k2 r ∂r rj l k2 r rb ε3 j l k3 r ∂r∂ ryl k2 r − ε2 yl k2 r ∂r∂ rj l k3 r ; n3 n2 ∂r∂ ryl k2 rj l k2 r yl k2 r ∂r∂ rj l k2 r

15

rb

ω BE;shell l;m

E;shell ∂ n2 ∂r∂ rj l k3 r − n2 ∂r∂ rhl k3 rAE;sca ω l;m ω − n3 ∂r rj l k2 rAl;m n3 ∂r∂ ryl k2 r

:

(16)

rb

B. Calculation of the Second Harmonic Sources The second step of our model is the determination of the nonlinear polarization oscillating at the SH frequency. As discussed in the introduction, SHG is forbidden in the bulk of centrosymmetric media within the dipolar approximation. Nevertheless, this symmetry is broken at the interface between two centrosymmetric media, thus allowing for SHG. Besides, it is known that the nonlinear polarization is also arising

2ω P s-sm surf;⊥

∞ X l X

C s-sm l;m Y l;m θ; ϕ;

(20)

l0 m−l

where the C l;m coefficients for both interfaces can be calculated using the spherical harmonics properties. After the re-incorporation of the expressions obtained for fundamental electric fields, these coefficients can be written as

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C c-s l;m χ ⊥⊥⊥

×

Butet et al.

X p l1 l1 1l2 l2 1 l1 ;m1 1 l2 ;m2 1

X X ASH;core 2ω l;m ∇ × j l K 1 rXl;m ; K1 l m0;2

Eshell 2ω

X X ASH;shell 2ω l;m ∇ × j l K 2 rXl;m K2 l m0;2

E;shell AE;shell l1 ;m j l1 k2 a Bl1 ;m yl1 k2 a k2 a

E;shell AE;shell l ;m j l2 k2 a Bl2 ;m yl2 k2 a × 2 k2 a Z × dΩY l;m Y l1 ;m1 Y l2 ;m2 ;

C s-sm l;m χ ⊥⊥⊥

×

(21)

X p l1 l1 1l2 l2 1 l1 ;m1 1 l2 ;m2 1

E;shell AE;shell l1 ;m j l1 k2 b Bl1 ;m yl1 k2 b k2 b

E;shell AE;shell l2 ;m j l2 k2 b Bl2 ;m yl2 k2 b k2 b Z × dΩY l;m Y l1 ;m1 Y l2 ;m2 :

×

(22)

dΩY l;m Y l1 ;m1 Y l2 ;m2

r 2l 12l1 12l2 1 4π l l1 l2 l l1 l2 × : m m1 m2 0 0 0

Eshell;∥ 2ω − Ecore;∥ 2ω −

l l1 l2

even;

m m1 m2 :

(25)

(26)

Since m1 and m2 can only take the 1 values (see above, Part A), only the modes for which m 0, 2 are required to write the two nonlinear polarizations. Having determined the expressions for the two SH polarization sheets, the SH electric field can be written everywhere in space. C. Calculation of the Second Harmonic Field The final step is the determination of the electric fields oscillating at the SH frequency in all space, namely within the nanoshell but also in the surrounding medium. The scattered electric field and the electric fields inside the nanocore and the nanoshell therefore can be written as Esca 2ω

X X ASH;sca 2ω l;m ∇ × hl K 3 rXl;m ; K3 l m0;2

(27)

(29)

(30)

(31)

at the core-shell interface and Esca;∥ 2ω − Eshell;∥ 2ω −

Hence, the following selection rules are immediately deduced from the Wigner 3j symbols’ properties: (24)

4π ∇ P c-s 2ω; ε1 2ω ∥ surf;⊥

Hshell;∥ 2ω − Hcore;∥ 2ω 0;

(23)

l ≤ l1 l2 ;

BSH;shell 2ω l;m ∇ × yl K 2 rXl;m ; K2

(28)

where K 1 n1 2ωk2ω, K 2 n2 2ωk2ω and K 3 n3 2ωk2ω with k2ω 2ω ∕ c. Similar to the case of the fundamental fields, the expansion coefficients for the harmonic electric fields are obtained applying the boundary conditions at both interfaces. The set of boundary conditions for a nonlinear polarization sheet standing at an interface has already been given by T. Heinz [47]. Considering a nonlinear polarization normal to the interface, it can easily be shown that only the modes with components of the electric fields normal to the interface can be excited. The boundary conditions for the SH fields are thus expressed as

All C l;m coefficients then vanish unless the integral R dΩY l;m Y l1 ;m1 Y l2 ;m2 is non-zero. This integral can be expressed as a product of Wigner 3j symbols [39]: Z

Ecore 2ω

4π ∇ P s-sm 2ω; ε3 2ω ∥ surf;⊥

Hsca;∥ 2ω − Hshell;∥ 2ω 0;

(32)

(33)

at the shell-surrounding medium interface, where ∥ denotes the component parallel to the interface. For each contributing mode, the boundary conditions lead to the following set of equations: ε1 2ω

j l K 1 rASH;core 2ω l;m K 1r

j l K 2 rASH;shell 2ω yl K 2 rBSH;shell 2ω l;m l;m ; (34) ε2 2ω K 2r ra

j l K 2 rASH;shell 2ω yl K 2 rBSH;shell 2ω l;m l;m K 2r hl K 3 rASH;sca 2ω l;m ; ε3 2ω K 3r rb

ε2 2ω

p 4πi ll1C c-s ASH;core 2ω ∂r∂ rj l K 1 r l;m − l;m K1 ε1 2ω

(35)

ASH;shell 2ω ∂r∂ rj l K 2 rBDH;shell 2ω ∂r∂ ryl K 2 r l;m l;m ; (36) K2 ra

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p 4πi ll1C s-sm ASH;sca 2ω ∂r∂ rj l K 3 r l;m l;m K3 ε3 2ω −

ASH;shell 2ω ∂r∂ rj l K 2 rBSH;shell 2ω ∂r∂ ryl K 2 r l;m l;m : (37) K2 rb

Since the spherical harmonics vectors are orthogonal, the excited modes are those for which the C l;m coefficients do not vanish. For this reason, the multipolar modes involved in the description of the SHG response from a core-shell nanostructure and from a homogeneous nanosphere are identical. The difference lies in the expansion coefficient expressions, i.e., the weight of each mode. Indeed, the nanoshell material properties only depend on the distance from the origin located at the center of the nanocore, not on the angles θ; φ. Both nanostructures possess the spherical symmetry. For each orthogonal mode, the four equations above allow the determination of the four unknown coefficients and their calculation is straightforward. As for the case of the linear electric fields, these coefficients give the SH electric field inside and outside the metallic nanoshell. Nevertheless, we will focus our attention on the SH scattered field which is the one experimentally observable. The corresponding coefficient is given by ASH;sca 2ω l;m

p T 4 C c-s 4πi ll 1 T 6 C s-sm l;m l;m − T 6 T 3 T 5 T 4 ε3 2ω ε1 2ω

(38)

with T1

ε2 2ωj l K 2 r ∂r∂ rj l K 1 r − ε1 2ωj l K 1 r ∂r∂ rj l K 2 r ; ε1 2ωK 2 2ωj l K 1 r ra (39)

T2

ε2 2ωyl K 2 r ∂r∂ rj l K 1 r − ε1 2ωj l K 1 r ∂r∂ ryl K 2 r ; ε1 2ωK 2 j l K 1 r ra (40)

ε2 2ωj l K 2 r ∂r∂ rj l K 3 r − ε3 2ωhl K 3 r ∂r∂ rj l K 2 r T3 ; ε2 2ωj l K 2 rK 3 rb (41) T4

j l K 2 r ∂r∂ ryl K 2 r − yl K 2 r ∂r∂ rj l K 2 r ; K2 rb

(42)

T5

n2 ε3 2ωhl K 3 b T ; n3 ε2 2ωj l K 2 b 1

(43)

T 6 −yl K 2 bT 1 j l K 2 bT 2 :

(44)

It is interesting to compare Eq. (38) with the expression obtained for the scattering coefficient in the case of SHG from a solid sphere. In order to perform this comparison, we consider a homogeneous sphere, i.e., a nanoshell with a shell and a core having identical properties. In this specific case, the interface

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between the core and the shell, and therefore the nonlinear polarization P c-s 2ω 0 associated, vanish. Since the shell surf;⊥ and core dielectric constants are similar, then T 1 T 5 0 and Eq. (38) reduces to 2ω ASH;sca l;m

p 4πi ll 1 C s-sm l;m : T3 ε3 2ω

(45)

As expected, this expression is identical to the one obtained in the case of the SHG from a solid homogeneous nanosphere [33,36]. This reduction shows the equivalence between our calculations and existing models. A rigorous validation of our model will be done in the near future by comparison with experimental measurements.

3. DISCUSSION In this part, the theory developed in the previous section is used to describe the SH response from silver and gold nanoshells beyond the dipolar approximation. We consider metallic nanoshells with an empty core, i.e., n1 1, dispersed in water, i.e., n3 1.33. A detailed analysis of the effect of the refractive index of the dielectric core and the embedding medium on the SHG from plasmonic nanoshells will be reported elsewhere. The dielectric constants for silver and gold are taken from [48]. The size effect is introduced for both silver and gold [5]. Indeed, the electron scattering at metal interfaces, in comparison to the corresponding metal bulk volume, increases for nanoparticles smaller than the electron mean free path, thereby modifying the surface plasmon lifetime and broadening the resonances [49]. In all our calculations, because of the thin nanoshells investigated, this effect is taken into account at both the fundamental and harmonic frequencies, using a surface scattering parameter A 1, a reduced electron mean free path d b − a introduced as the shell thickness and a Fermi velocity vf 1.4 106 m ∕ s taken with an identical value for gold and silver. The frequency dependence of the nonlinear susceptibility is also considered using a χ ⊥⊥⊥ component calculated using the following equation [44]: χ ⊥⊥⊥ −

aRS eε ε ω − 1 02 ; 4 r mω

(46)

where e and m denote the charge and the mass of the electron respectively, ε0 εr ω ε2 ω, and aRS is the so-called Rudnick-and-Stern parameter [50]. It is related to the harmonic nonlinear surface currents driven by the fundamental field in a direction normal to the interface. In a recent work, we have shown that in the case of silver nanoparticles this parameter is dispersion free as long as no interband transitions are involved at both the fundamental and harmonic frequencies [20]. The impact of these interband transitions on the Rudnick and Stern parameters is still unknown and the SHG intensity will be reported as relative values, the aRS parameter being fixed to unity in the present work. This hypothesis neither alters the relative weight of the dipolar and quadrupolar modes for a given fundamental wavelength nor their surface Plasmon resonance positions. The impact of these interband transitions on the Rudnick and Stern parameter can be readily introduced in the model. Finally, it is reminded that the interband transitions threshold is 3.85 eV (322 nm) in silver and 1.85 eV

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(670 nm) in gold. As in the case of the SHG from a metallic nanosphere, the total SH radiated power is an incoherent sum of contributions from the different contributing modes, even if interference between modes are revealed recording the SH wave in particular emission directions [18,51]: C SH sca 2ω

c

∞ X

8πK 23 2ω l;m

jASH;sca 2ωj2 : l;m

(47)

The contribution of the dipolar and quadrupolar modes to the total SH scattering cross section are calculated using l 1 and l 2, respectively. The octupolar (l 3) contribution to the total SH scattering cross section has been also calculated but has been found negligible for the nanoshell dimensions discussed in this work [18]. Note finally that we do not consider any breaking of the centrosymmetry through shape defects, for example, as could be the case in real nanoshells. Finally, the linear extinction cross section is also calculated for a direct comparison between linear and nonlinear optical properties. A. Silver Nanoshells The linear optical properties of silver nanoshells are discussed first. The calculations of the extinction cross section for a silver nanoshell (a 20 nm, b 30 nm) have been performed and two resonances are clearly visible. The resonance occurring at longer wavelength (close to 470 nm) corresponds to wavelength close to the dipolar surface plasmon resonance whereas the resonance occurring at shorter wavelength (close to 400 nm) corresponds to wavelength close to the quadrupolar plasmon resonance. The calculation of the SH scattering cross section for the same silver nanoshell (a 20 nm, b 30 nm) has been performed for a fundamental wavelength range from 350 nm up to 1200 nm. The results are depicted in Fig. 2(b). The SHG spectrum is dominated by four surface plasmon resonances corresponding to the enhancement of the dipolar emission mode only, the quadrupolar emission mode only, and the enhancement of both. The difference between both spectra is explained as follows. The most efficient mechanism leading to dipolar SH emission is the E 1 E 2 → E 1 mechanism where the two terms on the left of the arrow refer to the nature of the interaction with the fundamental wave and the third term describes the SH emission mode [33]. Hence, this notation describes here a dipolar emission (E 1 ) coming from the combination of an electric dipole (E1 ) and electric quadrupole (E 2 ) excitation. The pure dipolar mode E 1 E 1 → E 1 mechanism is forbidden for centrosymmetric objects, like the nanoshells discussed in this work, since Eq. (25) is not satisfied. SH dipolar emission needs retardation effects at the fundamental frequency, corresponding at the leading order to the quadrupolar mode, to be allowed. On the contrary, the quadrupolar SH emission mode can be excited without retardation effects at the fundamental frequency and arises from the E1 E 1 → E2 mechanism. Indeed, the retardation effects are included in the quadrupolar mode of the SH emission. The resonances labeled 3 and 4 in Fig. 2(b) correspond to wavelengths for which the harmonic frequency is resonant with the quadrupolar and with the dipolar bright mode, respectively. Only the mode with symmetry properties identical to the surface plasmon resonance is enhanced, e.g., the SH dipolar (quadrupolar) emission mode is enhanced

Fig. 2. (a) Extinction cross section as a function of the wavelength for a silver nanoshell (a 20 nm, b 30 nm). The dipolar and quadrupolar resonances are labeled 1 and 2, respectively; (b) Total SH scattering cross section calculated for the same silver nanoshell and for the dipolar (dashed line) and quadrupolar (dotted line) emission modes. Resonances are labeled from 1 to 4 (see main text for details).

when the harmonic frequency is tuned close to the dipolar (quadrupolar) surface plasmon resonance. On the other hand, both dipolar and quadrupolar emission modes are enhanced at resonances 1 and 2. In these cases, the fundamental wavelength is tuned close to the dipolar and quadrupolar resonances and both emission modes take advantage of the high fundamental electric field. It is interesting to note that the quadrupolar resonance labeled 1 in Fig. 2(b) arises from the E2 E 2 → E2 excitation channel as required by Eq. (25). As discussed in the introduction, the optical response of metallic nanoshells can be controlled by the ratio x between the inner radius a and the outer radius b. A calculation has thus been performed for different core sizes keeping the shell thickness constant at a value of 10 nm. Figure 3 shows the total SH scattering cross section for an inner radius equals to 10 nm, 20 nm, and 30 nm. As the core diameter increases, the coupling between the surfaces plasmons on the inner and the outer surfaces increases and all the resonances corresponding to symmetric coupling shift toward longer wavelengths, see arrows in Fig. 3. The position of the symmetric dipolar plasmon resonance shifts faster than the position of the quadrupolar one in this range of aspect ratio, x ranging from 0.41 up to 0.67. B. Gold Nanoshells In this part, the case of gold nanoshells is discussed in order to address similarities and differences with the silver ones. The

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Fig. 3. Total SH scattering cross section calculated for silver nanoshells with the same shell thickness (10 nm) but for increasing core dimensions. The evolution of resonance positions is shown by the arrows.

Fig. 5. Total SH scattering cross section calculated for gold nanoshells with the same shell thickness (10 nm) but for increasing core dimensions. The evolution of resonance positions is shown by the arrows.

calculations of the linear extinction cross section for a gold nanoshell (a 30 nm, b 40 nm) have been performed and only one resonance is observed [Fig. 4(a)]. Contrary to the case of silver nanoshells, the contribution of the quadrupolar mode to the whole extinction cross section is found negligible. Indeed, the amplitude of the quadrupolar surface plasmon resonance is far smaller than that of the dipolar surface plasmon

resonance. Figure 4(b) shows the total SH scattering cross section from the same gold nanoshell (a 30 nm b 40 nm) for a fundamental wavelength ranging between 500 nm and 1500 nm. Retardation effects are larger for short fundamental wavelengths and the SH scattering cross section increases as the fundamental wavelength decreases. As in the case of silver nanoshells, four resonances are observed and their physical origin, i.e., the excitation channels involved, is identical to the one discussed previously. In contrast, their position, width, and amplitude are different. In the case of gold nanoshells, the amplitude to width ratio is lower than in the case of silver nanoshells. This characteristic is mainly due to the increase of the surface plasmon damping induced by the creation of electron-hole pairs via interband transitions [52]. The nonlinear optical response of gold nanoshells is also tuned by changing the ratio x of the inner to outer core diameters as shown in Fig. 5. As in the case of silver nanoshells, the coupling between the surfaces plasmons on the inner and the outer surfaces increases as the inner radius a increases, leading to a redshift of the resonances corresponding to a symmetric coupling (see arrows in Fig. 5).

4. CONCLUSIONS

Fig. 4. (a) Extinction cross section as a function of the wavelength for a gold nanoshell (a 30 nm, b 40 nm). Only the dipolar resonance is visible because the amplitude of the quadrupolar one is too small. (b) Total SH scattering cross section calculated for the same gold nanoshell and for the dipolar (dashed line) and quadrupolar (dotted line) emission modes. Resonances are labeled from 1 to 4 (see main text for details).

In conclusion, the nonlinear Mie theory was extended to describe the case of SHG in metallic nanoshells, therefore including retardation effects at both the fundamental and harmonic wavelengths. Our model was applied to the calculation of total SH scattering cross sections showing that the contributions of the dipolar and the quadrupolar SH emission modes have a similar weight. The multipolar expansions of the electric field turn out to be very useful for the determination of both the allowed excitation channels and the mechanism of the scattered intensity enhancement. The impact of the aspect ratio was then discussed along with the wavelength dependence on the SH scattering cross section for gold and silver nanoshells. The effect of the dielectric constants of the core and of the surrounding medium on the SHG cross section from nanoshells can now be readily addressed with this model. This should allow further investigations, currently in progress, to discuss practical applications like sensing similarly to what has been proposed for homogeneous nanospheres [53].

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