Understanding the Second-Harmonic Generation Enhancement and

Article pubs.acs.org/JPCC

Cite This: J. Phys. Chem. C 2018, 122, 15635−15645

Understanding the Second-Harmonic Generation Enhancement and Behavior in Metal Core−Dielectric Shell Nanoparticles Sergey A. Scherbak†,‡ and Andrey A. Lipovskii*,†,‡ †

St. Petersburg Academic University RAS, Khlopina Str., 8/3, St. Petersburg 194021 Russia Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya Str., 29, St. Petersburg 195251 Russia

‡

J. Phys. Chem. C 2018.122:15635-15645. Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 09/28/18. For personal use only.

S Supporting Information *

ABSTRACT: To explain recently observed up to 50-fold enhancement of the second harmonic (SH) signal from gold nanoparticles after depositing of a thin dielectric layer, we perform an analytical study of the SH generation by metal core−dielectric shell spherical nanoparticles (NPs). We derive analytical expressions for the SH intensity accounting for electric dipole and quadrupole impacts. On this basis, we discuss the SH response spectrum of coated gold and silver nanoparticles. The possibility of adjusting the SH scattering regime to dipole or quadrupole resonant mode by varying the shell thickness is shown. We demonstrate that the thin, up to several NPs radii, dielectric cover of the NPs gives a significant advantage in the SH intensity in a line with the resonant enhancement. This effect is most pronounced in gold nanoparticles and even dominates resonant enhancement of the SH generation. This new result in nonlinear plasmonics provides a simple recipe for gold nanoparticles: thicker dielectric coating−higher the second harmonic signal. Consideration of thicker (order of optical wavelength) coatings shows that spatial Mie resonances can result in additional several orders enhancement of the SH signal generated by metal core−dielectric shell spherical NPs. consideration consists of three subsequent steps. The first one is the classical (linear) Mie problem28to derive a distribution of fundamental fields throughout the space. Regarding core− shell nanospheres, a substantial work was published by Aden and Kerker in 1951,29 who first derived corresponding Mie coefficients. At the second step of the nonlinear theory, the calculated fundamental electric field gives a spatial distribution of nonlinear sources. At the third step, the resulting SH electromagnetic fields caused by these nonlinear sources are to be calculated. The classical works of Dadap and co-workers23,24 are of high importance. These authors derived analytical expressions for the second order nonlinear dipole and quadrupole moments of a spherical particle, albeit in a small particle limit. Nonetheless, they rigorously analyzed the influence of the spherical symmetry and selection rules on features of the SH generation and accurately considered impacts of all components of the surface second order susceptibility tensor in a line with a bulk contribution in SHG. Notations of mechanisms of optical second-order interactions introduced by them are in use in many following works. Afterward, this theory was extended to the case of an arbitrary sized particle accounting for higher multipole terms.26 Several experimental studies were dedicated to nonlinear properties of core−shell NPs where either core or shell is plasmonic, e.g., dielectric core−metal shell

1. INTRODUCTION The effect of the second-harmonic generation (SHG) was discovered in 19611 and theoretically described in 1962.2 Since then, it has been of great interest for both applied and fundamental physics. The effect is generally weak like almost all nonlinear optical phenomena, and very intense light-waves are required. Many efforts are aimed at searching ways to increase the nonlinear response. One of the approaches is based on using the surface plasmon resonance (SPR) phenomenon in metal nanoparticles (NPs).3,4 These NPs are known to greatly enhance an electric field of an incident light wave at the SPR that significantly strengthens a nonlinear response.5−12 Plasmonic properties of NPs are very sensitive to their composition/structure, surrounding environment, size, and shape13−17 that allows spectral positioning of the SPR. Tuning the SPR by coating metal NPs with differently thick dielectric layers was multiply demonstrated.18−21 Besides possibilities to tune resonant properties, a cover layer protects NPs from undesirable interaction with a surrounding. Control of the SPR in metal NPs by their covering with dielectric layer looks promising for nonlinear optics because coating thickness affects the nonlinear properties of a particle indirectly only via changes in local fields. In recent decades, many papers starting from one of the first classical works by Sipe and co-workers22 have been dedicated to nonlinear optical phenomena in the presence of plasmonic metal surfaces. Theories concerning SHG by metal NPs are usually based on the nonlinear Mie theory.23−27 Generally, the © 2018 American Chemical Society

Received: April 12, 2018 Revised: May 24, 2018 Published: June 5, 2018 15635

DOI: 10.1021/acs.jpcc.8b03485 J. Phys. Chem. C 2018, 122, 15635−15645

Article

The Journal of Physical Chemistry C spheres,30 Ag-coated LiNbO3 cuboids,31 KNbO3−Au core− shell nanowires,32 BaTiO3−Au core−shell nanostructures,33 third harmonic generation by Ag−SiO2 core−shell nanoparticles.34 Recently, we have studied SHG by gold nanoisland films on a glass substrate covered with a differently thick TiO2 layer.35 We initially intended to tune the particles resonant wavelength to the wavelength of the excitation to gain advantage in the SHG intensity. However, the experiment demonstrated monotonic growth of the SH signal with increasing the coating thickness independently on the particles SPR position. We explained this for the first time registered phenomenon using a brief estimation, which showed that both resonant and nonresonant local fields grew with the permittivity of the surrounding medium. It is essential that the impact of the nonresonant local fields dominated in the SHG, which provided the observed behavior of the SH signal intensity. However, more accurate explanation of the nonlinear processes in dielectric covered metal nanoparticles is required. Note that in the present study we consider spherical NPs for analytical clarity, and it is not the exact modeling of the NPs from ref 35, which were placed on a substrate, therefore, the spherical symmetry was broken. Nonetheless, the phenomena under discussion are general and the given model qualitatively valid for understanding experimental results from ref 35. This paper is aimed at establishing the influence of a dielectric cover on the SHG by spherical metal nanoparticles. Here, we theoretically study the SHG by metal core−dielectric shell plasmonic NPs. Since noble metals, which are of interest, are known to be centrosymmetric materials,36 we take into account only the surface nonlinear susceptibility, while the bulk contribution is out of interest. Considering derivation of the fundamental fields throughout the space we mostly follow Bohren and Huffman,37 whose methodology of the problem we find most friendly. Besides, in the Supporting Information, we present the full set of the expressions for the coefficients required to calculate the electromagnetic fields throughout the whole space. This was done by neither Bohren and Huffman37 nor Aden and Kerker,29 since they assumed only the scattering coefficients to be of interest and provided only corresponding expressions. Further, in detail we go through derivation of the analytical expressions for the second-order nonlinear response of spherical core−shell nanoparticles. It should be noted that the similar SHG problem for the inverted structure (dielectric core− metal shell NPs) has been already considered by Butet and coworkers theoretically27 and experimentally.30 The other theoretical work regarding SHG by dielectric core−metal shell NPs was published by Wunderlich and Peschel.38 Butet and coworkers27 mostly focused on the effect of the ratio inner/outer radii of the metallic shell on the SHG. However, this does not clarify the influence of the dielectric core itself, since the area of the nonlinear surfaces varies. Thus, the consideration of metallic nanoshells27,30,38 did not allow separating the effect of a dielectric core and the size effect. Wunderlich and Peschel38 also focused on the effect of a differently thick metal shell, including influence of Mie resonances in a dielectric core. However, the effect of a dielectric core on local fields was not discussed. In this study, we generally use the approach from ref 27, though the methodology27 cannot be applied directly since different electric field expansions (in the core, not in the shell) are used. In ref 38 explicit derivation is not presented, but nonetheless, it should be pointed out that we use the same basis spherical harmonics as in ref 38. Next, we analyze general features of the SHG by a

noncovered NP of Drude metal. Despite the features of the SHG by bare NPs were revealed and discussed by Dadap,23,24 simple and straightforward explanation of these features is required here to discuss the SHG by the core−shell structures. Further, we study and compare the influence of differently thick dielectric covers on the SH response of silver and gold NPs. We consider two types of this influence: caused by local-field effects and caused by wave effects, which are, essentially, spatial Mie resonances in the structure. The influence of local fields, when the coating thickness is no more than a few particle radii, directly relates to the phenomenon of significant nonresonant enhancement of the SH signal reported in our previous work35 whereas spatial Mie resonances can provide further, up to several orders, increase of the SH signal.

2. THEORY As discussed above, we solve the problem of the nonlinear (SH) scattering in three steps. In the first step, the fundamental fields throughout the space are derived. In the second step, we define the nonlinear sources via the calculated fundamental fields. The third step is the derivation of the nonlinear response generated by these nonlinear sources. The scheme of the problem under consideration is presented in Figure 1. X-polarized and Z-directed plane light-wave of

Figure 1. Scheme of the problem.

frequency ω (wavelength λ) illuminates a spherical particle (core) of radius a coated with a spherical layer (shell) of outer radius b and thickness h = b − a. k is the wave-vector of the incident wave; E0 is its electric field directed along X-axis; r, θ, and φ denote the standard spherical coordinates. Refractive indices of the core, the shell and the outer medium are, respectively, ncore, nshell, and nout (εcore, εshell, and εout are corresponding dielectric permittivities, which equal respective indices squared). Note that we consider nonmagnetic media, therefore, μ = 1 everywhere and it is omitted in all the following equations. I. Fundamental Fields. The Mie theory of light scattering by a spherical particle of an arbitrary radius28 was first extended to the case of a coated particle by Aden and Kerker in 1951.29 However, here and below we follow the notations used by Bohren and Huffman.37 The methodology of the problem given by them is one of the most straightforward and clear. We present only that part of the solution, which is essential to the further study. Nevertheless, in the Supporting Information, the full solution is given in details for the sake of consistency. In the current study, only the equation for the fundamental electric field inside the core should be considered. Its expansion by vector spherical harmonics (see the Supporting Information for the details) is 15636


Article

The Journal of Physical Chemistry C ∞

Ecore =

(1) ∑ En(cn M(1) o1n − idn N e1n) n=1

where En =

2n + 1 E0i n n(n + 1) ,

component (see the Supporting Information, eq S6). This allows writing the expression for E⊥(ω) at the NP surface

(1)

∞

E⊥ =

E0 is the the incident field magnitude

j (m1x)

∑ −Enidn n n=1

further taken as unit, n is a polar eigennumber, cn and dn are (1) dimensionless coefficients of the expansion, Mo1n and N(1) e1n are vector spherical harmonics, which explicit form is presented in the Supporting Information, eqs S4−S7. As shown below, we can use only the electric inner coefficient dn to derive the nonlinear sources. This coefficient is used in the expansion of the electromagnetic fields inside the core and is responsible for socalled electric resonances. The expression for dn is

m1x

cos(φ)Pn1(cos θ)n(n + 1)

where jn is a spherical Bessel function of the first kind, and Pmn (cos θ) is the m-th associated Legendre polynomial of the n-th order. Accordingly, surface second order polarization is proportional to squared eq 5: ∞ s P⊥s (2ω) = −χ⊥⊥⊥

∑

En1En2dn1dn2

jn (m1x)jn (m1x) 1

2

(m1x)2

n1, n2 = 1

1 dn = ψn′(m1x)

cos2(φ)Pn11(cos θ)Pn12(cos θ)n1(n1 + 1)n2(n2 + 1) im1[ψn′(m2x) − A nχn′(m2x)]

Ps⊥

At the same time,

m2ξn′(y)[ψn(m2y) − A nχn (m2y)] − ξn(y)[ψn′(m2y) − A nχn′(m2y)]

P⊥s (2ω) =

(2ω) can be expressed as

∑ ∑ Cmn cos(mφ)Pnm(cos θ)

(7)

n=1 m

where

1

m2ψn′(m1x)χn (m2x) − m1ψn(m1x)χn′(m2x)

y= m2 =

(3)

2πnoutb are dimensionless size parameters; λ nshell are relative refractive indices; ψn(ρ), nout

1

χn(ρ), ξn(ρ) are, respectively, Riccati−Bessel spherical functions of the first kind, Riccati−Bessel spherical functions of the second kind, and Riccati−Hankel spherical functions of the first kind (sometimes called Riccati−Bessel spherical functions of the third kind). II. Nonlinear Sources. Next, the nonlinear sources are to be defined via the fundamental fields calculated in the previous section. In the frames of the present study, we consider centrosymmetric materials, which noble metals are known to be.36 Therefore, the bulk second-order response is forbidden and only surface second order nonlinearity should be considered. In a general case, three independent non-zero components of the surface second-order susceptibility tensor exist, χs⊥⊥⊥, χs⊥∥∥, χs∥∥⊥, but it was shown that the contribution of χs∥∥⊥ in SHG by noble metals can be neglected.39,40 The methodology presented below allows calculating impact in SHG by both generally nonvanishing terms, χs⊥⊥⊥ and χs∥∥⊥. However, here for the sake of analytical clarity, we suppose that the surface second order nonlinearity is mainly driven by χs⊥⊥⊥ component of the second order susceptibility tensor. Influence of all the tensor components on nonlinear optical processes was discussed in the classical work of Dadap23,24 and its extension by Gonella.26 We also assume that the core−shell interface is the only nonlinear surface, i.e., nonlinearity of the outer shell surface is neglected. This is reasoned by our focus on nonlinearity caused by plasmonic effects. Therefore, SHG by the outer dielectric border, at which evanescent plasmonic-induced local fields are much weaker than directly at the metal core surface, is out of interest. Under these assumptions, we write the expression for the surface second-order polarization of the core−shell interface as s P⊥s (2ω) = χ⊥⊥⊥ E⊥(ω; r = a − 0)E⊥(ω; r = a − 0)

1

The well known relation cos2(φ) = 2 + 2 cos(2φ) and the orthogonality of the {cos(mφ)} set indicate that the only nonvanishing terms of m-summation in eq 7 are the ones with m = 0 and m = 2. Comparing expansions (6) and (7) and using orthogonality properties of Pmn (cos θ), one can find unknown coefficients Cmn using the equation:

m2ψn′(m1x)ψn(m2x) − m1ψn(m1x)ψn′(m2x)

2πnouta , λ ncore m1 = n , out

x=

(6)

∞

(2)

An =

(5)

Cmn

∫−1 Pnm(t )Pnm(t ) dt

1 s = − χ⊥⊥⊥ 2

∞

∑

n1(n1 + 1)n2(n2 + 1)En1En2dn1dn2

n1, n2 = 1

jn (m1x)jn (m1x) 1

2

2

1

∫−1 Pnm(t )Pn1 (t )Pn1 (t ) dt 1

(m1x)

2

(8)

The coefficients like C mn are usually called Gaunt coefficients41,42 and the integration in the right side of eq 8 can be expressed via Wigner 3j symbols:42 1

∫−1 Pnm(t )Pνμ(t )Ppm+μ(t ) dt (n + m) ! (ν + μ) ! (p + m + μ)! (n − m) ! (ν − μ) ! (p − m − μ)! p yz ij n ν p yzij n ν z jj zj j 0 0 0 zzjj m μ −m − μ zz k {k {

= 2·( −1)m + μ

(9)

Therefore 1

∫−1 Pn0(t )Pn1 (t )Pn1 (t ) dt

i n1 n2 n yzij n1 n2 n yz zzjj zz = −2 n1(n1 + 1)n2(n2 + 1) jjj k 0 0 0 {k 1 −1 0 { 1

2

(10) 1

∫−1 Pn2(t )Pn1 (t )Pn1 (t ) dt 1

2

=2 (n − 1)n(n + 1)(n + 2) n1(n1 + 1)n2(n2 + 1) ij n1 n2 n yzij n1 n2 n yz jj zzjj zz k 0 0 0 {k 1 1 −2 {

(4)

The field expansion of interest (in the core) is described by eq 1, and only Nemn has a radial, i.e., normal to the surface,

(11) 15637


∞

Integrals in the left side of eq 7 are 1

∫−1 Pn0(t )Pn0(t ) dt = 1

∫−1 Pn2(t )Pn2(t ) dt =

2 2n + 1

∇ P⊥s (2ω) (12)

2 (n − 1)n(n + 1)(n + 2) 2n + 1

From eqs 19−22, it is evident that the SH fields are driven by the SH polarization the same way as the fundamental fields are driven by the incident field (see the fundamental boundary conditions, eqs S16 and S17, in the Supporting Information). Therefore, the expansion in eq 23 dictates the expansion form of the SH fields. Because of the choice of the polarization expansion in eq 7, the angular dependence in eq 23 exactly coincides with the angular dependence of Nθ,φ emn (see eq S6 in the Supporting Information). Therefore, only this vector spherical harmonic should be used in the expansion of the SH electric fields and, consequently, only Memn − in the SH magnetic fields:

(13)

Thus, coefficients Cmn and, therefore, second order nonlinear polarizability can be found using eq 8 and eqs 9−13. Moreover, properties of 3j-symbols directly result in the following selection rules: n ≤ n1 + n2

(14)

n + n1 + n2 − even

(15)

Here the physical meaning of eigennumbers n, n1, n2 should be clarified. n1 and n2 indicate a fundamental multipolar mode (n1,2 = 1 − dipole, n1,2 = 2 − quadrupole, etc.), whereas n indicates the second harmonic multipolar mode born by fundamental modes n1 and n2. Further, we limit our analysis with dipole and quadrupole terms only. Note that the consideration of the quadrupole mode is essential, because, as follows from selection rules 14 and 15, the pure dipole regime (n1 = n2 = n = 1) is forbidden. Note that the selection rules here are the same as the ones for the SHG by a nanosphere in a homogeneous medium.23,24 In the notations introduced by Dadap, allowed mechanisms of second order interaction in this case are E1 + E2 → E1

(16)

E1 + E1 → E2

(17)

∞

Ecore(2ω) =

Hcore(2ω) = −

Hsc(2ω) =

θ ,φ θ ,φ (Eshell (2ω) − Esc (2ω)) = 0, r = b

(21)

θ ,φ θ ,φ (H shell (2ω) − H sc (2ω)) = 0, r = b

(22)

∑ ∑

2ω (1) dmn M emn(2ω)

(25)

n = 1 m = 0,2

∑ ∑

2ω (3) iamn N emn(2ω)

(26)

Koutc 2ω

∞

∑ ∑

2ω (3) amn M emn(2ω)

(27)

n = 1 m = 0,2 ∞

Eshell(2ω) = −i ∑

∑

2ω (1) 2ω (2) gmn N emn(2ω) + wmn N emn(2ω)

n = 1 m = 0,2

(28)

Hshell(2ω) = − +

K shellc 2ω

∞

∑ ∑

2ω (1) gmn M emn(2ω)

n = 1 m = 0,2

2ω (2) wmn M emn(2ω)

(29)

Kcore, Kshell, and Kout being SH wavenumbers in a corresponding medium. Note that here we use the second harmonic expansion 2ω 2ω 2ω coefficients d2ω mn, amn, gmn, wmn in the dimensional form in contrast to their fundamental analogues. Substitution eqs 24−29 into the boundary conditions, eqs 19−22, results in the system of equations: 2ω y m2ψn′(m1x) −m1ψn′(m2x) m1χn′(m2x)zyz ijj amn zz jij 0 z jj zz jjj jj zz jj 2ω zzzz jj 0 z −ψn(m2x) ψn(m1x) χn (m2x) zz jj dmn zz jj z zz·jjj jj zz jj 2ω zzzz jj m ξ ′(y) z ′ ′ 0 ψn(m2y) −χn (m2y) zz jj gmn zz jj 2 n z jj zz jjj jj z j 2ω zzzz j ξn(y) 0 −χn (m2y) zz jj wmn ψn(m2y) z k {k { ij 4πi m1m2x yz jj Cmn zz jj ε (2ω) a zz jj 2 zz jj zz j zz = jj 0 zz jj zz jjj zz 0 jj zz j z 0 (30) k {

(19) (20)

∞

Kcorec 2ω

n = 1 m = 0,2

4π ∇θ , φ P⊥s (2ω), r = a ε2(2ω)

θ ,φ θ ,φ (H shell (2ω) − Hcore (2ω)) = 0, r = a

(24)

∞

Esc(2ω) =

where E1 and E2 denote dipole and quadrupole modes, respectively, the left terms correspond to fundamental modes, which form nonlinear response, and the right term corresponds to the SH mode. III. Nonlinear Fields. Now we can derive nonlinear fields generated by this source using the surface nonlinear polarization given by eq 7. Ecore (2ω) and Hcore (2ω) are, respectively, electric and magnetic SH fields inside the core; Eshell (2ω) and Hshell(2ω), the SH fields in the shell; Esc (2ω) and Hsc (2ω), the SH fields in the outer medium. Boundary conditions in the presence of the nonlinear surface polarization are known:43 θ ,φ θ ,φ (Eshell (2ω) − Ecore (2ω)) = −

2ω (1) −idmn N emn(2ω)

∑ ∑ n = 1 m = 0,2

(18)

E2 + E2 → E2

ÄÅ ÅÅl dPnm(cos θ ) | o o1 = ∑ ∑ CmnÅÅÅÅm cos( m φ ) eθ̑ } o o o o Å dθ ÅÇn r ~ n=1 m ÉÑ l Pnm(cos θ ) | o ÑÑÑÑ o m +m − sin(mφ) }eφ̑ ÑÑ o o o sin θ o (23) n r ~ ÑÑÖ Article

The Journal of Physical Chemistry C

For the sake of generality, ε2 in the boundary condition in eq 19 denotes εout in the case of the noncoated particle and εshell in the other case. The gradient in the right part of eq 19 calculated via eq 7 is 15638


Article

The Journal of Physical Chemistry C Note that in eq 30 the size parameters x, y and the dispersive refractive indices m1,2 should be taken at frequency 2ω. Corresponding expansion coefficient obtained from eq 30 are 4πi m1x Cmn ψn′(m1x) m2ξn′(y)[ψn(m2y) − A nχn (m2y)] − ξn(y)[ψn′(m2y) − A nχn′(m2y)] ε2(2ω) a

4πi m1x Cmn ′ ′ ′ ′ ψn(m1x) m2ξn(y)[ψn(m2y) − A nχn (m2y)] − ξn(y)[ψn(m2y) − A nχn (m2y)] ε2(2ω) a

4πi m1x Cmn ψn′(m1x) m2ξn′(y)[ψn(m2y) − A nχn (m2y)] − ξn(y)[ψn′(m2y) − A nχn′(m2y)] ε2(2ω) a

ψn(m1x)

(34)

It follows from eqs 31−34 that the dimensional factor of the expansion coefficients is Cmn . Therefore, we can substitute

a C C 2ω Cmn 2ω 2ω Cmn 2ω 2ω → an a , dmn → dn a , gmn → wn2ω amn , → gn2ω amn , wmn 2ω 2ω 2ω where a2ω n , dn , gn , and wn are corresponding dimensionless

2ω amn

q0 =

factors in eqs 31−34. Factor An is given by eq 2 and should be calculated at frequency 2ω. The next is to derive the total SH intensity. For this, we integrate Poynting vector over a surface encircling the particle. We present here the resulting equation (see details in the Supporting Information): W2ω =

LE1(Ω) =

3/2

15εout(2ω)

9εshellεout 3

( ) (ε

(εshell + 2εout)(εcore + 2εshell) + 2

x y

core

LE2(Ω) =

2

25εshellεout 5

( ) (ε

2

εcore(2ω) ε2(2ω)

− εshell)(εshell − εout)

(39)

(2εshell + 3εout)(2εcore + 3εshell) + 6

x y

core

− εshell)(εshell − εout)

(40)

(35)

All the dielectric permittivities in eqs 39 and 40 should be taken at the corresponding frequency. Noteworthy that the equations obtained correspond ones presented by Dadap and coauthors24 with the only principal difference in local field factors LE1 and LE2, which here are the ones for a core−shell spherical structure. Also note that Dadap and coauthors employed convention EΩ(r,t) = 2Re[EΩ(r) exp(−iΩt)] for all the oscillating quantities. This convention was proposed by Pershan,44 and accordingly to Bloembergen45 it results in 4-fold difference in power compared with standard notation EΩ(r,t) = Re[EΩ(r) exp(−iΩt)] which we tacitly used. Therefore, in our equation for the SH intensity factor 4, which can be found in the work of Dadap,24 is absent. The final step is to introduce frequency dependence of the second order surface nonlinear susceptibility χs⊥⊥⊥ of noble metals:40,46 α eε s χ⊥⊥⊥ (ω) = − RS (εcore(ω) − 1) 02 (41) 4 mω

The form of eq 35 shows the general behavior of the SH intensity. Here |Cmn|2 indicates proportionality to the forth power of the enhancement factor (local field factor, Lω) of the 2 fundamental electric field (see eq 8), whereas |a2ω n | is essentially proportional to squared enhancement factor (local field factor, L2ω) of the SH electric field. Therefore, this corresponds to the known dependence:4 W2ω ∼ |L2ω|2|Lω|4 E40 or, qualitatively the same, W2ω ∼ |E2ω|2|Eω|4. Despite eq 35 is a general one and stays valid for arbitrary sizes of the core/shell, in many cases it can be more convenient to use small-particle approximation (m1,2x, m1,2y ≪ 1). Small parameter expansion of the spherical Bessel functions in eq 35 results in 4 cKout

(38)

where

ÅÄÅ ∑ ÅÅÅÅÅ2 2n(n + 1) |C0nan2ω|2 ÅÇ 2n + 1 n=1 Å 2

W2ω =

16π 3 2 s a E0 χ⊥⊥⊥ LE1(ω)LE1(ω)LE2(2ω) 5

∞

ÉÑ Ñ 2(n − 1)n (n + 1) (n + 2) 2ω 2 Ñ + |C2nan | ÑÑÑÑ ÑÑÖ 2n + 1 2 8a 2Kout

(33)

where p0 and q0 are, respectively, effective SH dipole and quadrupole moments. Neglecting the impact of the weak mechanism E2 + E2 → E2 we present the equations defining SH effective moments: 8πi s p0 = kouta3E02χ⊥⊥⊥ LE1(ω)LE2(ω)LE1(2ω) (37) 15

2ω 2ω − χn (m2x)wmn ψn(m2x)gmn

c εout(2ω)

(32)

[ψn′(m2y)ξn(y) − m2ψn(m2y)ξn′(y)][ψn′(m2x) − A nχn′(m2x)]

ψn(m1x)

2ω = wmn

(31)

[χn′(m2y)ξn(y) − m2χn (m2y)ξn′(y)][ψn′(m2x) − A nχn′(m2x)]

ψn(m1x)

2ω = gmn

2ω = dmn

[ψn′(m2x) − A nχn′(m2x)]

ψn(m1x)

2ω = amn

1 ij y jj5 |p0 |2 + Kout |iq0|2 zzz 6 k {

(36) 15639


Article

The Journal of Physical Chemistry C where e and m are charge and mass of electron, and αRS is socalled Rudnick−Stern parameter.46 However, all the frequencyindependent constants in eq 41 act as a kind of a general scale factor for the SH intensity. Also we follow the hydrodynamic model22 and neglect the frequency dependence of αRS. Therefore, the essential dependence is s χ⊥⊥⊥ (ω) ∼

εcore(ω) − 1 ω2

(42)

Impacts of the other components of χ̑s tensor also can be calculated using this methodology. However, it is necessary to remind that if Ps∥ (2ω) component appears, trivial boundary condition (20) changes to43 θ ,φ θ ,φ (H shell (2ω) − Hcore (2ω)) =

4π 2iω[eȓ × PS]θ , φ , r = a c (43)

3. RESULTS AND DISCUSSION We start the discussion with a description of general features of the SH response spectrum. Although these were derived and discussed by Dadap,23,24 the straightforward and simple explanation is definitely required for the aims of the present study. We feel necessity to give this clarification because the nature of the SH response features is far from obvious, and its understanding is essential to start the main discussion. First, consider the SH intensity generated by a spherical particle of 10 nm in radius without coating. The material of the particle is Drude metal with complex permittivity εcore(ω) = 1 +

Figure 2. (a) Optical absorption spectrum of a 10 nm in radius spherical particle of Drude metal (γ = 1014 s−1, λp = 300 nm) without coating. (b) Spectra of the total SH intensity and dipole and quadrupole impacts; the total intensity spectrum has an offset of 102 along y-axis. Insets: SH radiation patterns at the corresponding resonances.

iωp2 ωγ − iω 2

(44)

We chose relaxation rate γ = 1014 s−1 and plasma wavelength 2π c λp = 300 nm (plasma frequency ωp = λ ). The parameters of

For the particle which is small compared to the light wavelength, a single dipole resonance is observed in the linear absorption spectrum. Its wavelength is marked as λE1 = 522 nm in Figure 2a. Calculated quadrupole resonance, though it is almost invisible in the spectrum of the small particle, is marked at λE2 = 475 nm in Figure 2a. Nevertheless, in the SH spectrum (Figure 2b) four different resonances marked as 1−4 are seen. We start the discussion with long-wavelength resonances 1 (1044 nm) and 2 (950 nm), which are pure dipole and quadrupole. Evidently, these resonances occur at doubled dipole and quadrupole resonant wavelengths, 2λ E1 and 2λ E2 , respectively. They correspond to the cases when the SH field is resonantly enhanced at dipole (1) or quadrupole (2) resonance of the particle. The appearance of the quadrupole response in the SH spectrum, despite dipole linear regime, originates from the selection rules and allowed mechanisms of the SH interaction described by the expressions 16−18. The point is that the supposed-to-be-dominant pure dipole process E1 + E1 → E1 is forbidden by selection rules, and the quadrupole mode participates in all the allowed processes. The allowed mechanisms are listed in Table 1 with their relation to the resonances marked in Figure 2b. Also, here we introduced local field factors (LFFs) of dipole and quadrupole modes, LE1,E2, which reflect relations between incident and induced electric fields of corresponding modes. Expressions for the SH intensity in terms of LFFs for each mechanism of the SHG are listed in the third column of Table 1. Note that these expressions are written formally to indicate different impacts. Exact values of the SH intensity are calculated in accordance with eq 35.

p

the Drude metal, first of all λp, were chosen to make the Drude material similar to real noble metals in optical range. However, this resulted in high discrepancy between Drude and real metals in the near-infrared region. Making a Drude metal more coincidental in the infrared region otherwise causes some nonphysical issues at shorter wavelengths, e.g., fitting the Ag dispersion with the Drude model in infrared gives λp ≈ 129 nm, whereas experiments give essentially longer wavelength, λp ≈ 328 nm.47 We believe that the demonstration with the Drude metal with these parameters is more indicative and the exact infrared dispersion, which has no relation with plasmon resonances, is not crucial. The graphs illustrating the permittivity dispersion are presented in the Supporting Information. It is worth to note that the imaginary part of a Drude metal permittivity is a monotonous function of frequency, which weakly influences resonant properties of a nanoparticle. On the contrary, in real metals (see the Supporting Information) the nonmonotonic character of the imaginary part of the permittivity can dramatically affect both linear and nonlinear spectral properties as discussed below. The optical absorption spectrum calculated in accordance with eq S31 of the Supporting Information and the spectrum of the SH response are presented in Figure 2. In line with the total SHG intensity, dipole and quadrupole impacts in the SHG are plotted separately. All of the calculations here and below were carried out for the incident wave of a unit amplitude. 15640


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for a small particle. Note that the SH radiation is polarized correspondingly to the dipole or quadrupole kind of radiation, and this was discussed in details by Dadap.23,24 In addition, the drop in the SH intensity at doubled plasma wavelength, 2λp, should be pointed. At λp wavelength the permittivity of Drude metal is approximately zero (precisely equals if a lossless metal with γ = 0 is considered). That results in trivial boundary condition for the SH fields and, therefore, in the SH intensity drop. However, exciting and intriguing SH properties of a bare metal nanoparticle are not the main goal of this paper, which is aimed at the influence of a coating dielectric layer. Generally, plasmonic NPs with a dielectric shell inherit properties of noncovered ones. That is why we have presented the discussion above. However, consideration of a coating layer brings many new interesting results. We consider the SH intensity spectra of coated NPs made of noble metalstypical plasmonic materialssilver and gold. Dispersion of the dielectric permittivity of copper is very close to one of gold, therefore, the results for a copper nanosphere are about the same and not discussed separately. The metals dispersions here and below are from Johnson and Christy data47 and they are presented in Figure S2 in the Supporting Information. To demonstrate the general behavior of the coated nanoparticles, here we consider a permittivity of the shell εshell = 5.5 that is very close to TiO2, and further, we call the coating material TiO2, despite the real dispersion is not accounted for. Note that all the equations presented here allow substituting dispersive coating permittivity for accurate calculations. In Figure 3, we present the SH intensity spectra of 10 nm in radius Ag NP covered with differently thick TiO2 layers. Corresponding optical absorption spectra are in the inset in Figure 3.

Table 1. Description of SHG Resonances resonance no. 1 2 3d 3q 4d 4q

mechanisms of SH interactiona E1 (nonresonant) + E2 (nonresonant) → E1(resonant) E1 (nonresonant) + E1 (nonresonant) → E2(resonant) E1 (resonant) + E2 (nonresonant) → E1(nonresonant) E1 (resonant) + E1 (resonant) → E2(nonresonant) E1 (nonresonant) + E2 (resonant) → E1(nonresonant) E2 (resonant) + E2 (resonant) → E2(nonresonant)

impacts in SHGb W2ω sc ∼ |LE1 (ω)LE2 LE1 (2ω)|2 W2ω sc ∼ |LE1 (ω)LE1 LE2 (2ω)|2 W2ω sc ∼ |LE1 (ω)LE2 LE1 (2ω)|2 W2ω sc ∼ |LE1 (ω)LE1 LE2 (2ω)|2 W2ω sc ∼ |LE1 (ω)LE2 LE1 (2ω)|2 W2ω sc ∼ |LE2 (ω)LE2 LE2 (2ω)|2

(ω) (ω) (ω) (ω) (ω) (ω)

a

E1 and E2: dipole and quadrupole modes, resonant modes are marked with bold. Left side of the expressions: fundamental modes (ω). Right side: SH modes (2ω). bLE1,E2: factors of the local field enhancement for dipole and quadrupole modes, respectively.

The SHG processes listed in the second column of Table 1 should be interpreted in a way that in the SH output all impacts are multiplied, e.g., expression for resonance 1 could be schematically written as [SH response] ∼ [nonresonant dipole response at fundamental wavelength] × [nonresonant quadrupole response at fundamental wavelength] × [resonant dipole response at SH wavelength]. This point is indicated in the third column of Table 1. Continuing the discussion of resonances 1 and 2, it follows for resonance 2 that the quadrupole SH mode originates from two fundamental dipole modes, nonresonant but nonetheless strong, whereas the dipole SH mode (resonance 1) originates from the strong dipole mode and the weak quadrupole one. That is why the resonances 1 and 2 are almost of the same order of intensity (quadrupole one is even half an order stronger). It is notable that the resonances of this kind were distinguished experimentally in silver colloids.48 The situation is almost the same with resonances 3 and 4. Only the fundamental modes are resonantly enhanced at these wavelengths. Besides, the SHG regimes at these resonances are not pure dipole and quadrupole ones but contain both impacts, which ratio strongly depends on the particle radius. Quadrupole and dipole impacts in resonances 3 and 4 are denoted as 3q and 3d; 4q and 4d, respectively. In resonance 3, the quadrupole contribution 3q is dominant. Comparing mechanisms for 3q and 3d in Table 1, we see that 3q contains two resonant multipliers, whereas for 3d there is only one. That explains the dominance of 3q. Likewise, resonance 4q is barely seen (a slight shoulder in the spectrum) comparing to 4d. Although 4q has two resonant multipliers vs one for 4d, all its multipliers are quadrupole, and size parameter smallness suppresses the resonant enhancement. For larger particles the SHG regimes at resonances 3 and 4 differ from almost pure dipole/quadrupole. However, it is interesting standalone phenomenon that a small particle at dipole resonance scatters the SH wave mostly in the quadrupole regime (maximum 3 in Figure 2b) and, on the contrary, at quadrupole resonance in the dipole regime (maximum 4 in Figure 2b). The SH radiation patterns, i.e., angular dependencies of the SH intensity, at resonances 1−4 are presented in the insets in Figure 2b near the corresponding resonant peaks. The SH patterns are rotated relatively their fundamental analogues; e.g., the effective SH dipole momentum is directed along the direction of the fundamental wave propagation kvector rather than along E-vector. As was discussed above, the patterns of the resonances 1, 4 and 2, 3 are almost the same only

Figure 3. SH intensity spectra of 10 nm in radius Ag NP covered with TiO2 layer (εshell = 5.5) of different thickness labeled near each curve. The curves are artificially separated along y-axis with the offset of 103. Inset: linear optical absorption spectra of the same structure.

In these spectra, both the SH and optical absorption, we observe the red shift of all the resonances with the coating thickness increase. This red shift has a saturation behaviorat a certain thickness the local fields are almost completely localized in the coating layer and further increase in thickness barely affects plasmonic properties of the particle. This is a well-known phenomenon which was multiply discussed.18−21 Also, general growth of the absorption magnitude is evident. However, its nonmonotonic behavior is caused by the nonmonotonic 15641


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The Journal of Physical Chemistry C behavior of the dispersion of imaginary part of Ag dielectric permittivity (see the Supporting Information). The resonances in the SH spectra in Figure 3 are marked in accordance with the notations introduced in Figure 2b and in Table 1, e.g., at resonance 1 the SH wave is resonantly enhanced in dipole regime, and the corresponding mechanism is listed in the first row in Table 1, and so on. It is interesting to note that at some thicknesses the dipole and quadrupole resonances are merged (1 + 2 and 3 + 4) and resolved again only for thicker coatings; i.e., we observe two separate resonances in the spectrum of the bare particle, they are merged in the cases of thin, 2 and 5 nm, coating (marked as 1 + 2 and 3 + 4) and resolved again starting from 10 nm coating. This is explained by Figure 4 where the resonant

Figure 5. Left axis: SH intensity of TiO2-covered Ag nanoparticle 10 nm in radius vs coating thickness at the wavelength of 930 nm (solid line). Right axis: linear electric field in the center of the full-dielectric (TiO2) nanosphere of radius h + 10 nm at wavelengths of 465 nm (dashed) and 930 nm (dotted); the curves are normalized by corresponding fields at h = 0.

resonances 1 and 2 in Figure 3. These peaks, 1 and 2, are caused by local-field effects, when the field is not fully localized in the coating layer, and the layer thickness strongly influences the particle plasmonic properties. Moreover, besides shift of the resonances, the dielectric coating causes general growth of local fields (both at the resonance maximum and at the tails) and, therefore, intensity of the SH signal grows regardless the resonance position as was demonstrated in the experiments.35 This effect arises from the fundamental point that local fields increase with increasing the permittivity of an outer medium εout in accordance with the well-known formula3 3εout E loc ∼ E0 εMe + 2εout (45)

Figure 4. Dipole λE1 and quadrupole λE2 resonant wavelengths vs coating thickness. Inset: Difference between dipole and quadrupole resonance wavelengths vs coating thickness.

wavelength vs coating thickness is shown. Quadrupole fields are more localized than dipole ones, and therefore, the coating thickness affects them first. This results in the nonmonotonic dependence of the difference Δλ between dipole and quadrupole resonant wavelengths on coating thickness, which is presented in the inset in Figure 4. At thicknesses for which Δλ is small (a few nanometers), the resonances are not resolved separately. Moreover, the quadrupole resonance wavelength can even be longer than a dipole one (Δλ < 0) at some thicknesses. However, for a thick coating, in which the both fields are localized, the resonances are well-split, and the dependences saturate. Above, it is shown that the plasmonic resonant wavelengths can be tuned in a wide range (see Figure 3) via a dielectric coating. It is also useful to consider the dependence of the SH intensity on coating thickness at a certain wavelength rather than the full spectrum. This dependence shows that at thicker coatings (a few hundred of nm) spatial Mie resonances, which strongly affect the SHG, appear in a line with plasmon resonances. In Figure 5 we present the dependence of the SHG intensity on the coating thickness at the fundamental wavelength of 930 nm (the wavelength was chosen for demonstration purposes). The dependence in Figure 5 demonstrates a significant growth (several orders of magnitude) of the SHG efficiency with coating thickness. In Figure 5, four features marked as 1, 2, I, and II can be distinguished. Note that types of resonances and SHG regimes of the coated particle here are inherited from ones of a bare NP, which were discussed above and presented in Figure 2 and Table 1. Peak 1 corresponds to coating thickness at which the SH dipole resonant wavelength coincides with the excitation wavelength of 930 nm. The same is peak 2, but the resonance is quadrupole. Peaks 1 and 2 in Figure 5 match, respectively,

where εout in the numerator acts as a kind of a scale factor. This scaling appears independently on resonant effects, which are driven by the denominator. The same reasoning is legit in the case of a cover of a finite thickness. Such covering layer can be treated as an effective outer medium with permittivity between εout and εshell, which monotonically grows with coating thickness and saturates at the level of εshell. Furthermore, as follows from eq 35 and as was discussed above, the SH intensity W2ω ∼ |E2ω|2|Eω|4; i.e., it is proportional to the second power of the SH field and to the forth power of the fundamental field. In the case of about 1 μm excitation wavelengths, which is under discussion, the SH field is typically near-resonant and the fundamental field is far from resonances. Thus, peaks 1 and 2 in Figure 5 correspond to the resonances of E2ω, whereas the difference marked by Δ in the inset in Figure 5 is caused by both the nonresonant (tail) increase of Eω in accordance with eq 45 and by proximity of the resonance of E2ω. The effect of nonresonant local field increase is much more pronounced and evident in gold NPs which are discussed below. Also note that in case of a small particle the field localization and, therefore, characteristic shell thicknesses scale with the particle radius. In contrast to these local-field effects, peaks I and II arise because of spatial resonances. This is illustrated in Figure 5 by the dependences of electric fields in the center of a full-TiO2 nanosphere with radius h+10 nm at wavelengths of 465 and 930 15642


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The Journal of Physical Chemistry C nm. Thus, peak I corresponds to the case when the SH wave has a spatial resonance in the core−shell structure. At the thickness corresponding to peak II, both the fundamental wave and the SH wave resonate, the SH spatial resonance being of the next order. Generally, dispersion of the coating material should separate these resonances, i.e., split peak II into two. However, these resonances are broad because of low quality-factor of the structure, and they hardly can be well-split even in the case of a highly dispersive shell. Resonances of the electric field at 465 nm are slightly shifted from the peaks I and II of the SH intensity, since this field is calculated in the near-field region, whereas the SH intensity is in the far-field region. The discrepancy between near-field and far-field resonances is natural.49 At the same time, the peak of |E|λ=930 nm and peak II of the SH intensity match accurately because just the fundamental electric field inside the core drives the SHG. In addition, these dependences illustrate that the electric field can be up to 4−5-fold enhanced due to spatial Mie resonances in the dielectric sphere. Let us make a qualitative estimation in accordance with the relation W2ω ∼ |E2ω|2|Eω|4 at cover thickness of 275 nm (peak II region): W2ω ∼ |3.0|2|5.5|4 ∼ 104 above the level of so-called local-field enhancement. This estimation, albeit very rough, indicates that 104−105 boost of the SH intensity caused by the spatial Mie resonances should not surprise. Note that these resonances, I and II, depend on the wavelength, shell thickness and its index, rather than on radius and plasmonic properties of the metal core if it is small (a ≪ h). The SH signal enhancement of this kind can be essential the same way as in a SH-tuned laser resonator with a frequency doubling crystal inside. It should be also noted that varying the thickness of the coating, which is thick enough to fully localize plasmon-induced fields, does not directly affect plasmonic properties of the core−shell structure, but nonetheless allows tuning the SHG via spatial Mie resonances. Note that the case when a fundamental wave spatially resonates in a dielectric core resulting in enhancement of the SHG by a dielectric core−metal shell structure, was discussed by Wunderlich and Peschel.38 Next, we discuss the results obtained for a gold nanosphere. Note that the dependences for a copper nanoparticle are almost the same. These two materials have less vivid plasmon resonances than silver, but nonetheless are of interest for nonlinear plasmonics. In Figure 6, the SH spectra for 10 nm in radius Au nanoparticle covered with differently thick TiO2 shell are presented. The dependences presented in Figure 6 are qualitatively similar to ones for a silver particle in Figure 3 and the same way the red shift of the resonances is observed. However, the resonant curves for Au particles are broader than ones for Ag, since gold is more lossy. Therefore, the quadrupole and dipole resonances in Figure 6 are barely resolved, except the case of the thickest covers. Once more, we note that the general properties of covered nanoparticles are essentially the same as of bare ones. The sequence and nature of the resonances in Figure 6, except their merging, correspond ones introduced in Figure 2 and Table 1. Therefore, the resonances in Figure 6 are not marked to avoid overloading of the figure. Contrary to the silver nanosphere, a significant growth in resonance magnitude with the gold nanosphere cover thickness is seen in Figure 6. One of the reasons is the character of Au dispersion. The cover shifts the resonance to longer wavelengths where the imaginary part of gold dielectric permittivity (i.e., losses) decreases from 2.6 at 520 nm to 1.0 at 660 nm (see the Supporting Information). One more reason is the nonresonant

Figure 6. SH intensity spectra of 10 nm in radius Au nanoparticle covered with differently thick TiO2 layer (εshell = 5.5). Inset: corresponding linear absorption spectra.

growth of local fields with the shell thickness, which was discussed above. Furthermore, because of this nonresonant local field growth each spectrum with a thicker coating almost completely lies above ones with thinner coating. We purposefully do not separate the curves in Figure 6 along yaxis, as we did in Figure 3 for the silver NP to make the effect under discussion more evident. It should be also noted that the SH spectra of a bare nanoparticle lie above the spectra of the nanoparticle with a thin coating, e.g., 2 nm. In the cases of “nocoating” and “any-coating” the properties of the particle surface, which is nonlinear, are changed threshold-like, since particlesurrounding dielectric contrast is changed (see eq 19 and the comment below it). Note that this difference is caused only by the effect of the SH boundary conditions, the possible influence of the dielectric surrounding on χs⊥⊥⊥ is not accounted for. Growth of the resonance magnitude in the silver nanoparticle in Figure 3 is less evident than in the gold one. This is because the dispersion of the imaginary part of silver dielectric permittivity is not monotonous in that wavelength area (see the Supporting Information), and this can suppress the effect of local fields increase. In Figure 7 we present the dependence of the SH intensity by 10 nm in radius gold nanoparticle at standard wavelength of Nd:YAG-lasers, 1064 nm, vs coating thickness. The dependence in Figure 7 is qualitatively similar to the one for a silver nanosphere (Figure 5). The same way two regimes of the

Figure 7. SH intensity of a 10 nm in radius TiO2-covered Au nanoparticle vs the coating thickness at wavelength 1064 nm. 15643



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enhancement of the SH signal can be distinguished. The first is a thin coating case when the signal growth is due to the enhancement of the local fields caused by coating. This case is of particular interest, since in core−shell NPs sizes of a core and a shell are typically of the same order. Importantly, the dependence for the gold nanoparticle has no resonant features at thin coatings, which were observed for silver one (marks 1 and 2 in Figure 5). Thus, in gold nanoparticles the effect of the nonresonant growth of Eω caused by the coating dominates the resonance effects of E2ω. By itself, this is a new result for nonlinear nanoplasmonics−covering a gold nanoparticle with a dielectric leads to stronger SHG independently on resonance position. Further increase in the coating thickness results in the SH growth associated with spatial Mie resonances of the fundamental or the SH wave in the core−shell nanostructure. This phenomenon is not directly related with plasmonic properties of the particle as was mentioned above. Therefore, these parts of the dependences for Ag and Au nanoparticles are almost the same. Note that there is no strict border between these regimes, and the separating dashed line in Figure 7 is no more than a guide for eyes.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b03485. Equations of the linear Mie theory for core−shell nanospheres; derivation of the scattering intensity of the SH wave; plots of dispersions of Drude metal used and silver- and gold-based on experimental data47 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Tel: +7-812-4488591. Fax: +7-812- 4486998. E-mail: [email protected]. ORCID

Sergey A. Scherbak: 0000-0002-0507-5621 Andrey A. Lipovskii: 0000-0001-9472-9190 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors appreciate the support by the Ministry of Education and Science of the Russian Federation (Project No. 3.2869.2017).

4. CONCLUSION Finally, we analyzed the influence of the dielectric cover on the SHG by metal core−dielectric shell spherical nanoparticles. The expressions for all the expansion coefficients to calculate electromagnetic fields in the linear scattering problem are presented in the Supporting Information as a supplemental result. Analytical expressions describing the SHG by the core− shell spherical nanoparticles and straightforward explanation of the general features of this process was given. We considered dipole and quadrupole impacts in the SHG and discussed the nature and features of four resonances appearing in the nonlinear scattering spectrum following refs 23 and 24 in the case of noncoated NPs. After consideration of the SHG by a bare nanoparticle of Drude metal, spectral dependences of the SH intensity scattered by silver and gold nanoparticles covered with differently thick dielectric layer were modeled and discussed. The possibility of tuning SPR wavelength in a wide range, which allows gaining significant advantage in the SHG was demonstrated. It was shown that SHG regime (dipole or quadrupole) can be adjusted at a certain wavelength by varying the cover thickness. We described two kinds of influence of the dielectric cover on the SHG features. In the case of the thin coating (less than a few particle radii), the local field is not fully concentrated in the shell. In this mode, increasing the cover thickness red-shifts the SPR position, and local fields generally grow. This growth, especially in the case of gold nanoparticles, turned out to dominate the resonant enhancement of the field and caused 1.5−2.0 order increase in the SH signal in spite of detuning the SPR from the second harmonic wavelength. This results in a straightforward recipe for gold nanoparticles: thicker coating−higher SH signal. The other type of the influence appears in the case of a thicker coating (comparable with the wavelength in the shell material) and relates to wave-effects, contrary to the local-field effects. In such dielectric shell either the fundamental or the SH wave can spatially resonate at a certain thickness. This results in additional 4−5 orders enhancement of the SH signal and offers a way to select certain SH plasmonic mode via adjusting coating thickness to the spatial resonance of this mode. Essentially, varying the thickness of these thick coatings does not anymore affect the SPR wavelength.

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REFERENCES

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