1. Introduction

The absorption and scattering of light by small particles is a subject of relevance in several fields of science and technology and constitutes a problem of long-standing interest [1]. With the recent development of nano-optics, the importance of understanding and manipulating the optical response of nano-scale structures has become even more evident. Since it is well-known that metallic nanoparticles support electric-type resonances, a very common strategy for achieving this manipulation is based on the exploitation of the localized surface plasmon resonances (LSPR) of metallic nanoparticles [2,3]. These nanoresonators, which can be spectrally tuned by modifying the shape of the particle, provide some of the building blocks for the design of optical metamaterials [4].

A key ingredient in the search of metamaterials with novel optical properties is the ability to excite resonances of a magnetic character. In this respect, the split ring resonators (SRRs) proposed by Pendry et al. [5] constitute an important development in the field, and have motivated much of the work on the subject. When combined with structures that support electric-type resonances, SRRs can be used in the design of artificial materials with unusual optical properties. More recently, however, the realization that high-index dielectric nanoparticles can support strong magnetic-type resonances [6–12], has opened new alternatives for the design optical metamaterials. Interestingly, this kind of excitation occurs even for relatively simple structures like spheres or infinite cylinders, for which analytic solution to the scattering problem exist and have been known for over a century [13]. In addition, these structures present lower losses that their metallic counterparts.

Given the resonant nature of the metamaterial designs, it is clear that the internal fields can be quite strong and it is then not surprising that the possibility of exciting nonlinear optical effects in their constituents have also attracted some attention. Of particular interest to us here is the nonlinear generation of second harmonic radiation. The case of metallic nanoparticles, in which the excitations are mainly of dipolar electric origin, has been a well studied subject [14–17]. The role of dipolar magnetic resonances in the SHG by metallic structures and, in particular, by arrays of U-shaped nanoparticles (the optical version of the SRRs) has also been investigated by several groups, and it has been reported that such arrays can enhance the SHG efficiency with respect to those of other noncentrosymmetric particles. Nevertheless, the physical origin of the enhancement has been a point of some discussion in the literature. The initial studies seemed to indicate that the enhancement was linked to resonances of a magnetic-dipole type [18], but later studies that compare the response of arrays of magnetic split-ring resonators and arrays of complementary split-ring resonators (electric response) seem to indicate that in general there was no correlation between SHG efficiency and the excitation of the magnetic-dipole moment [19]. More recently, it was shown that the reported enhancement of the second harmonic in split-ring resonator based media is driven by the electric rather than the magnetic properties of the structure [20].

The SHG by dielectric particles that support magnetic-type resonances has received less attention, but is also a subject of interest and, as could be expected, the role played by the magnetic-type resonances has been one of the issues of interest. Most of these works deal with particles made on non-centrosymmetric materials and, in particular, AlGaAs and GaAs [21–25]. In a recent publication Kruk et al. [24] studied the role played by the excitation of multipoles in the SHG by nanodisks made from noncentrosymmetric materials, and found that both, electric and magnetic resonances were important for the nonlinear response. More recently, Smirnova at al. [26] reported studies of SHG by high-index dielectric nanoparticles made of centrosymmetric materials, and discussed the contributions of the different nonlinear sources and multipolar excitations in the generated field.

In this paper, we study the SHG by small two-dimensional particles made of centrosymmetric high index dielectrics in a spectral region in which strong magnetic dipole resonances can be excited. Since the method of calculation has been explained in some detail in previous work [27,28], we only present enough material to make the paper self-contained and facilitate the discussion. The calculation of the surface sources and the scattered fields at ω and 2ω are quite rigorous, and the nonlinear response of the medium is modeled on the basis the dipollium model [29]. Both, the linear and second harmonic fields are decomposed into their multipolar components, allowing the visualization of the elementary contributions to the second harmonic field, as well as those that drive the nonlinear processes.

2. Theoretical methods

The physical situation considered is illustrated in Fig. 1. A two-dimensional particle defined by the boundary between vacuum (medium I) and a nonlinear material (II) is illuminated by a p-polarized time-harmonic plane wave of frequency ω propagating in the direction of the x₁ axis. Medium II is assumed to be isotropic, homogeneous, and nonmagnetic and is characterized by its permittivity ϵ(ω). Apart from the condition of invariance along x₂, the shape of the particle can be rather arbitrary, and the curve describing its profile can be conveniently represented in terms of two functions of a parameter t. These two functions represent the x₁ and x₃ coordinates of the vector-valued function

 

Fig. 1 Schematic diagram of the geometry considered.

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The particle shapes employed in the calculations were generated using Gielis superformula [30,31].

The solution to the problem of scattering in the SHG depends on the solution of the linear problem and on the non-linearities of the medium, represented by the non-linear bulk polarization and the nonlinear surface susceptibilities. Since the medium composing the particle is assumed isotropic, most of the contribution to SHG will arise from nonlinear sources located on the surface of the particle. The procedure employed to calculate the scattered fields is fairly rigorous and based on the numerical solution of coupled surface integral techniques that determine the source functions for the fundamental and second harmonic fields [27]. As we have already mentioned, we only present the results that are needed for the discussion, focusing on the aspects that differ from the treatment presented in [27], which was formulated to deal with the case of metallic particles.

2.1. The nonlinear sources

For isotropic media the nonlinear polarization takes the general form [32]

Due to the invariance of the system along x₂, the scattering problem is essentially of a scalar nature and, at both, the ω and 2ω frequencies, there are two independent modes of polarization (s and p). With our assumptions, illumination with purely s or p waves produces only p polarization at the second harmonic frequency and, in order to produce an s-polarized field at 2ω, a combination of s and p waves would be required [27]. To keep the paper to a manageable size, we consider only the case of p-polarized illumination. The second harmonic signal produced under s-polarized illumination is at least two orders of magnitude smaller than that the one obtained with p polarization.

We will denote by ψ^(I)(r, θ|Ω) the x₂ component of the magnetic field $H_2^{(I)}(r,\theta |\Omega )$ in medium I, where, Ω stands for the fundamental (ω) or second-harmonic (2ω) frequencies. The fields are assumed to have a time dependence of the form exp(−iΩt), but the explicit reference to it will be suppressed.

The nonlinear boundary conditions satisfied by the (p-polarized) harmonic field can be written in the general form [27]

Starting from the expressions found in [27], and after some algebra, we obtain the following expressions for the nonlinear sources,

As we have already mentioned, we are interested in the case of high refractive index dielectric particles and, in particular, silicon. The required nonlinear susceptibilities and the bulk respond are calculated with the so-called dipollium model [29], which models the medium as a collection of polarizable entities with a single resonant frequency that are driven by the field. The permittivity of such a medium may be written as ϵ(ω) = 1 + 4πn_Bχ(ω), where n_B is the density of the polarizable entities, and the linear susceptibility is of the form

With this model, the frequency-dependent parameters α(ω), β(ω), and γ(ω) that enter into the general expression for the bulk nonlinear polarization [27] can be written in terms of the linear permittivity as [29]

The main difference between the free-electron model (FEM) adopted in [27] and the dipollium model employed here, is that the FEM assumes a linear susceptibility of the form χ(ω) = −e²/mω², and that the number density is that of the free electrons instead of that of the dipole oscillators in the material. It is then clear that the dipollium model reduces to the FEM when Γ and ω₀ ≪ ω.

With Eqs. (5), (7) and (8), we can determine the nonlinear response of the material in terms of the linear permittivity ϵ(Ω). We point out that the constant factor (n_Be)⁻¹ is common to all the nonlinear bulk and surface contributions [Eqs. (7) and (8)], and the second harmonic scattered field ${\psi }_{sc}^{(I)}(r,\theta |2\omega )$ will be scaled with this factor.

At this stage, it is worth saying a few words about the relative strengths of the second harmonic contributions arising from the bulk and the surface. For plasmonic structures, it is commonly assumed that the bulk contribution contribution is much smaller than that due to the surface. The case of dielectrics has only been addressed recently [34], with the conclusion that the surface and bulk effects are comparable. As Eqs. (3)–(5) show, the nonlinear sources have contributions from both, the bulk and the surface, and by switching off selectively one or the other, it is possible to assess their relative importance for the results. In the calculations presented here, the bulk contribution was three orders of magnitude lower than the surface contribution.

Inspecting the nonlinear sources of Eq. (4), we observe that the last term of Eq. (4b) represents the generation of a p-polarized signal with an s-polarized excitation. As we have already mentioned, in our calculations this p-polarized signal is two orders of magnitude smaller than the one obtained under p-polarized illumination.

2.2. The scattered field and the multipolar decomposition

For large values of r, the scattered field can be written as [27]

The source functions ${\psi }_p^{(I)}(t|\Omega )$ and ${\Upsilon }_p^{(I)}(t|\Omega )$ are calculated by solving numerically a pair of coupled integral equations and imposing the linear and nonlinear boundary conditions [27].

Outside a virtual cylinder that contains entirely the scattering particle, the field can be expressed in terms of cylindrical harmonics [35]

Following [36], we find it convenient to rewrite Eq. (11) as

When considering the expansion Eq. (12) at the fundamental frequency, we will denote the coefficients $c_m^{\pm }$ by $a_m^{\pm }$, while those at the second harmonic frequency will be denoted by $b_m^{\pm }$.

We note that the base ${\psi }_m^{+}(r,\theta ,|\Omega )$ contains functions that are symmetric about the direction of incidence, whereas the base ${\psi }_m^{-}(r,\theta ,|\Omega )$ contains functions that are antisymmetric about that direction. In other words, Eq. (12) is a representation of the field as the sum of an even and an odd function and, in general, the two contributions are needed to describe it completely [36]. It can also be observed that, for any order m, the ± contributions are rotated by an angle π/2m between them. The first few terms of the even and odd bases are illustrated in Fig. 2. It is also worth pointing out that, for the case of p polarization, the first three coefficients represent the magnetic dipole (MD, term with c₀), the electric dipole (ED, terms with $c_1^{\pm }$), and the electric quadrupole contributions (EQ, terms with $c_2^{\pm }$).

 

Fig. 2 Illustration of the lowest order functions of the symmetric and antisymmetric bases. In the symmetric base the lowest order terms represent the magnetic dipole (MD), the electric dipole (ED) and the electric quadrupole (EQ) contributions. In the antisymmetric base they are the electric dipole (ED) and the electric quadrupole (EQ) contributions.

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The strength of the multipolar contributions to the field scattered by a particle with non-circular cross sections, can be determined by evaluating its overlap with the base functions. Thus, the coefficients of the expansion can be calculated from the expression

Based on the far field approximation Eq. (9), for m ≠ 0 the overlap integral reduces to the simple expression

It is important to mention, however, that the higher order multipoles decay more rapidly than the lower order ones and, therefore, that their strength can only be determined from near field data. In consequence, the coefficients of the high order multipoles cannot be estimated from far field data.

2.3. The scattering efficiency

The total power scattered by a two-dimensional particle can be evaluated by constructing an imaginary cylinder of length L and radius r₀ that encloses the particle (see Fig. 1). Then, at the fundamental frequency the scattering efficiency of the particle can be expressed as [27]

The scattering efficiency at 2ω can be defined as

It is worth pointing out that in contrast with the linear case, in which the scattering efficiency has no dimensions, by its definition, the second-harmonic scattering efficiency has units of area over power (e. g. cm²/Watt).

It is straightforward to show that in terms of the expansion coefficients of Eq. (12), the scattering efficiency can be written as [1,28]

3. SHG by symmetric particles

With reference to Fig. 1, we shall say that a particle is symmetric if it is symmetric about the plane x₁ − x₂. It is clear that, for such particles, the linear source functions and the linearly scattered field must also be symmetric about the x₁ − x₂ plane; that is to say that ${\psi }_{sc}^{(I)}(r,\theta |\omega )={\psi }_{sc}^{(I)}(r,2\pi -\theta |\omega )$. A necessary implication of this symmetry is that the coefficients $a_m^{-}=0$ and therefore, that only the base ${\psi }_m^{+}(r,\theta |\omega )$ is needed for the representation of the field.

Similarly, at the second harmonic frequency, the field will be antisymmetric about the direction of incidence. This can be deduced from the expressions for the source functions given in Eq. (4) (the derivative of an even function is an odd function) or by invoking some simple physical arguments compatible with the assumed symmetry. With reference to Fig. 3(a), let us assume momentarily that the field scattered at 2ω is symmetric about the x₁ − x₂ plane. We allow now to pass the time and consider the situation after half a cycle for frequency ω (a full cycle at 2ω). The situation is illustrated on the left panel of Fig. 3(a); we see that the sign of the field at ω has changed, but that of the field at 2ω is unchanged. If we now flip the lower part of Fig. 3(a) about the x₁ − x₂ plane, we would end up with a situation that is different from the original one (the sign of the 2ω field would be different from the one assumed). This means that the assumption of a symmetric scattered field at 2ω led us to some unphysical or contradictory situation. On the other hand, is one assumes that the scattered field at 2ω is antisymmetric, as on Fig. 3(b), the same arguments lead to a physically compatible situation with the one assumed originally. Then, we have that ${\psi }_{sc}^{(I)}(r,\theta |2\omega )=-{\psi }_{sc}^{(I)}(r,2\pi -\theta |2\omega )$, and the coefficients $b_m^{+}=0$, so that only the base ${\psi }_m^{-}(r,\theta |2\omega )$ is needed to represent the scattered field. An immediate conclusion that can be obtained from this observation is that, for symmetric particles, $b_0^{+}=0$ and, thus, that even when the particles support a strong magnetic resonance at 2ω, this is not excited through the nonlinear process.

 

Fig. 3 Illustration of situations in which the second harmonic scattered field is assumed to be symmetric (a) and antisymmetric (b).

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To substantiate these arguments, we consider the specific example of SHG from a silicon cylinder (2D particle of circular cross section) of radius a = 150 nm. The case of the cylinder is particularly interesting, as it also allows us to test the approach by comparing the numerical results with those obtained with the analytical solution [27]. The wavelength of incidence spans the range from 800 to 3000 nm, so that the second harmonic covers the visible and near infrared range, from 400 to 1500 nm. As in all cases, the incident and second-harmonic waves are p-polarized.

The linear scattering cross section is shown in Fig. 4, together with the first three multipolar contributions which, due to the symmetry of the system are from the symmetric base ${\psi }_m^{+}(r,\theta |2\omega )$. The magnetic dipole contribution, denoted by the green dashed curve that peaks around λ = 1403 nm, is quite strong. The electric dipole contribution, shown with the dot-dashed red curve is also quite strong and peaks around λ = 977 nm. The quadrupole contribution is small in the range considered.

 

Fig. 4 Scattering efficiency at the fundamental frequency ω for an infinite cylinder (2D particle with a circular cross section) of radius a = 150 nm. The first three multipolar contributions are also shown. Note that due to the symmetry of the system, only multipoles of the base ${\psi }_m^{+}$ are excited in the linear case.

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Results for the second harmonic efficiency are shown in Fig. 5(a), together with the first three contributions from the multipole expansion. To help in the interpretation of the results, in Fig. 5(b), we present the results of the linear scattering efficiency in the same spectral region, together with the curves for the first 4 multipolar contributions. This figure is helpful to identify the kind of multipoles supported by the particle in this wavelength range and the position of their resonances. It is clear that the first three components of the even base are excited quite strongly and, thus, that the system supports well the magnetic dipole and the electric dipole and quadrupole modes. The $a_3^{+}$ term is weakly excited, but has a resonance around λ = 600 nm that somehow drives a relatively strong response in the second harmonic results (term proportional to $b_3^{-}$).

 

Fig. 5 Nonlinear and linear scattering efficiency of the cylinder considered in Fig. 4. (a) Generated second harmonic and the first three multipolar contributions when the fundamental wavelength λ spans the range shown in Fig. 4. Note that due to the symmetry of the system only multipoles of the base ${\psi }_m^{-}$ are excited. (b) Linear scattering efficiency in the same spectral region as in (a). In this case, only multipoles of the base ${\psi }_m^{+}$ are excited. In both cases, λ denotes the wavelength of incidence.

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The second harmonic efficiency has, as expected, no contributions from the magnetic dipole term b₀ and the excited multipoles belong to the antisymmetric base ${\psi }_m^{-}(r,\theta |2\omega )$. Although the system has a strong magnetic dipole resonance around λ = 1403 nm, the second harmonic sources are not able to excite it. The particle has also a strong electric dipole-type resonance around λ = 977 nm. However, this resonance (term proportional to $b_1^{-}$) is not excited strongly in the second harmonic. Clearly, the strongest contribution to the second harmonic efficiency is that corresponding to the electric quadrupole (coefficient $b_2^{-}$, denoted by the short-dashed blue curve). The scattering patterns at the wavelengths of resonance of the three contributions (shown as insets) confirm the predominance of the quadrupole term of the antisymmetric base (see Fig. 2). Also, since the contributions to the field are of the form $b_m^{-}$ sin mθ, no radiation is produced in the exact forward and backscattering directions (θ = 0°and 180°).

A near field map corresponding to $|{\psi }_{sc}^{(I)}(r,\theta |2\omega )|$ at the wavelength λ/2 = 977 nm, which coincides with the peak of the electric dipole resonance, is presented in Fig. 6. It is interesting to note that even at that wavelength, the quadrupole term seems to dominate the near field pattern.

 

Fig. 6 Modulus of the magnetic field in the near field of the cylinder considered in Fig. 4 at λ/2 = 977 nm, which coincides with the maximum of the electric dipole resonance in the second harmonic field. Note that even at the peak of the dipolar resonance, the quadrupole contribution is predominant.

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3.1. The effect of the fundamental multipoles in SHG

The effect of exciting multipoles at the fundamental on the SHG can be studied from the analytical expression of the scattering efficiency of the cylinder at 2ω [28]. The strength of the multipoles excited at the second harmonic, represented by the coefficients $b_m^{-}$, depend on the strength of the multipolar contributions to the fundamental frequency, represented by $a_m^{+}$. The effect produced in the second harmonic by each elementary excitation at the fundamental frequency can be visualized by artificially turning on and off particular multipolar contributions.

To illustrate the procedure and study these effects we continue with our example of a cylindrical silicon particle of radius a = 150 nm. In Fig. 7, we show the second harmonic efficiency, together with curves obtained when only one or two of the multipoles excited at the fundamental frequency are considered. The curve with the green dashed line corresponds to the case in which only the dipolar term $(a_1^{+})$ is kept at the fundamental, to drive the nonlinear processes that give rise to the second harmonic field. One can see that the strong quadrupolar peak around 726 nm is already present at the second harmonic with this fundamental dipolar excitation. On the other hand, if we keep all the excitations, except the dipolar one (grey curve in the figure), practically all the second harmonic response is lost. Interestingly, the nonlinear interaction between the dipolar and the quadrupolar excitations produces the second harmonic peaks that correspond to the $b_1^{-}$ and $b_3^{-}$ contributions.

 

Fig. 7 Scattering efficiency in the SHG for the cylinder considered in Fig. 4, considering the effect of the fundamental multipolar excitations. The full curve includes all the multipolar contributions at the fundamental. The green dashed curve considers only the dipolar electric contribution, the gray continuous line considers all the contributions but the dipolar electric one, and the short-dashed blue curve considers the dipolar and quadrupolar electric contributions.

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From the results presented in these figures we conclude that, for cylindrical particles, the magnetic dipole excitation at ω plays no role in the generation of second harmonic light, and that the magnetic dipole resonance supported by the particle at 2ω is not excited by the nonlinear processes involved. Since the MD mode is associated with a circulation of the electric field that is tangential to the surface of the cylinder, this observation is a consequence of the fact that, from the assumed model of the nonlinear polarization, the surface susceptibility ${\chi }_{ztt}^{\mathrm{s}}=0$.

4. Particles with non-circular cross sections

The results presented up to this point correspond to particles with a circular cross section, for which one can proceed by means of the analytical expressions for the coefficients of the multipolar expansions [28]. For particles with more general cross sections, the scattered fields are calculated numerically [27] and, from these results, the coefficients of the multipolar expansion are estimated from Eqs. (15)–(17). In this section, we present results for particles with non-circular cross sections and discuss the consequences of the lack of symmetry of the particle about the direction of incidence.

4.1. Particles with a rectangular cross section

Let us consider now a particle with a smoothed-square cross section whose side D = 300 nm. Results for the second harmonic efficiency are shown in Fig. 8. As expected, since the particle is symmetric about the x₁ − x₂ plane, only the components of the antisymmetric base are excited at the second harmonic frequency. In comparison with the case of the circular cylinder, we observe that with the square cross section particle the electric dipole excitation becomes relatively stronger. From the figure we observe that the scattering efficiency around the wavelength of the electric dipole resonance (∼ 1052 nm) is larger than in the other case, and that the scattering pattern seems to have a clear dipolar shape. On the other hand, in the position of the quadrupolar resonance, the other contributions are negligible and, in consequence, the scattering distribution is clearly quadrupolar in character. A map of |ψ_sc(r, θ)| at λ/2 = 1052 nm, which coincides with the maximum of the dipolar excitation is shown in Fig. 9. As expected, the near field pattern is dominated by the dipolar contribution.

 

Fig. 8 Second harmonic scattering efficiency for a dielectric particle with a 300 nm per side square cross section. Also shown are the first three multipolar contributions to the efficiency. The insets illustrate the form of the far field scattering distributions at the wavelengths of the three multipolar resonances.

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Fig. 9 Modulus of the magnetic field in the near field of the particle with square cross section considered in Fig. 8 at λ/2 = 1052 nm, which coincides with the maximum of the electric dipole resonance at the second harmonic wavelength.

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4.2. Non-symmetric particles

When the particle is not symmetric about the x₁ − x₂ plane it is possible to excite the two multipolar bases. We now consider particle whose cross section has the shape of an equilateral triangle, illuminated under different orientations. The sides of the triangle have a length of 285 nm and the illumination has a wavelength of λ = 1251 nm, which was chosen because the second harmonic wavelength (λ/2 = 626 nm) coincides with the quadrupolar resonance of the particle.

To illustrate the situation, in Fig. 10 we present the near field intensity maps corresponding to the four orientations considered. The angle α denotes the angle of rotation from the original orientation, shown in (a), the upper left image of the figure. When α = 0, the figure is symmetric about the x₁ − x₂ plane and it is expected that only the antisymmetric base will be excited. Since the field is antisymmetric, its squared modulus has to be symmetric about the x₁ − x₂ plane, and this property can be clearly observed in the figure. Further verification of the antisymmetric property of the field is provided by the calculated values of the expansion coefficients, which are shown in Table 1. We see that for α = 0°, the coefficients of the symmetric base are identically zero.

 

Fig. 10 Modulus of the magnetic field in the near field of the particle with triangular cross section with different orientations at λ/2 = 626 nm, which coincides with the maximum of the electric quadrupole resonance at the second harmonic wavelength. The symmetric orientation of the particle is shown in (a), while (b), (c) and (d) correspond to rotations by α = 10°, 20° and 30°.

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Table 1. Values of the multipolar expansion coefficients of the second harmonic field for different orientations of the particle of triangular cross section (see Fig. 10) at λ/2 = 626 nm. For simplicity, the common factor 1/(n_Be) has been omitted from the values shown.

View Table

Continuing with the discussion of Fig. 10(a), we observe that the generated near field pattern appear to be fairly close to what one would expect for a field dominated by a quadrupolar resonance, which coincides with the fact that the coefficient $b_2^{-}$ shown in the table for that case is the largest. Since the second harmonic wavelength coincides with the position of the quadrupolar resonance, this is not surprising. As the triangular particle is rotated, the near field pattern changes and we observe on the table that the contributions from the symmetric base become non-zero and grow with the rotation angle. Similarly the field along the x₃ = 0 line is no longer zero.

5. Summary and conclusions

We have studied the scattering and SHG of light by dielectric particles made of centrosymmetric a high index material, focusing on the case of silicon. The solution of the scattering problem at both, ω and 2ω is based on the numerical solution of coupled integral equations for the surface sources and the nonlinear response of the surface is modeled with the dipollium model. The scattered fields at ω and 2ω are written in terms of a multipolar basis that permits the visualization of the kind of excitations that take place at the particle.

Using simple symmetry arguments, we show that for the case of particles that are symmetric about an axis in the direction of incidence, the magnetic dipole contribution cannot be excited at the second harmonic frequency. This result is illustrated by calculations with particles with circular, square and triangular cross sections. Similarly, for the case of infinite cylinders (circular cross section), we also find that the magnetic dipole excitation at the fundamental frequency plays no role in the second harmonic generated by the particle, which is mainly driven by the electric dipole excited at the fundamental frequency. At the harmonic frequency, the response is normally dominated by the quadrupolar excitation.