Second-harmonic generation in Mie-resonant dielectric nanoparticles made of noncentrosymmetric materials
We develop a multipolar theory of second-harmonic generation (SHG) by dielectric nanoparticles made of noncentrosymmetric materials with bulk quadratic nonlinearity. We specifically analyze two regimes of optical excitation: illumination by a plane wave and single-mode excitation, when the laser pump drives the magnetic dipole mode only. Considering two classes of nonlinear crystalline solids (dielectric perovskite material and III-V semiconductor), we apply a symmetry approach to derive selection rules for the multipolar composition of the nonlinear radiation. The developed description can be used for design of efficient nonlinear optical nanoantennas with reconfigurable radiation characteristics.
pacs:42.65.РІв‚¬вЂ™k, 78.35.+c, 42.70.Nq
The resonant response is one of the main routes to increase the efficiency of nonlinear signal generation at the subwavelength scales in the absence of phase matching effects. That is why optical nonlinearity at the nanoscale is usually associated with the enhancement of electric fields in plasmonic nanostructures due to geometric plasmon resonances [1, 2]. Despite the significant progress in this area , there exist fundamental drawbacks that limit the efficiency of nonlinear generation with metallic structures. Besides the evident problem of high ohmic losses, typical metals have cubic lattice with inversion symmetry which restricts second-order nonlinear effects, such as the second-harmonic generation (SHG) . It can be observed only due to the surface effects or the field gradients in the bulk of nanoparticles [3, 5], which are relatively weak. Recently, a novel nanophotonic platform based on high-index dielectric nanoparticles has emerged . All-dielectric nanostructures are free from high ohmic losses, and offer wide variety of dielectric and semiconductor materials including those with nonzero bulk second order susceptibility tensor. Excitation of Mie resonances in such nanoparticles provides novel opportunities for nonlinear optics [7, 8], and allows one to achieve record-high nonlinear conversion efficiencies at the nanoscale [9, 10, 11, 12, 13, 14, 15].
Despite the intense experimental stuides of the SHG effects in Mie-resonant nanostructures, a comprehensive theory of the SHG emission from nanoparticles with nonzero bulk nonlinearity tensor has not been proposed yet. The important works related to the SHG generation were focused on the surface and bulk effects in nanoparticles with centrosymmetric crystalline lattice: in noble metal nanoparticles [16, 17, 18] including the shape effects , and in Mie-resonant silicon nanoparticles [20, 21]. In this work, we theoretically study the SHG by individual spherical high-index dielectric nanoparticles made of non-centrosymmetric materials (aluminum gallium arsenide AlGaAs and barium titanate BaTiO), which possess a large bulk quadratic susceptibility. These materials are actively employed for nonlinear all-dielectric nanophotonics [9, 22, 23, 24, 11]. We systematically describe the SHG in nanoparticles and mechanisms of its resonant enhancement, depending on the symmetry of the crystalline structure and polarization of the incident light. We employ methods of multipolar electrodynamics providing a transparent interpretation for the measurable far-field characteristics, such as radiation efficiency and radiation patterns [7, 25, 21].
Using analytical techniques, we demonstrate the ability to manipulate the nonlinear radiation of a spherical nanoparticle by varying illumination properties. By means of symmetry analysis of the SHG process we obtain the selection rules for the nonlinear generation, and identifiy which channels of multipole composition are active in SHG. These rules previously were known only for nanoparticles of a spherical  and arbitrary shape  made of centrosymmetric materials. The knowledge of these basic mechanisms of nonlinear generation in a single spherical nanoparticle can be extended in application to complex nonlinear structures, such as nanoparticle oligomers [26, 27] or nanoparticle arrays in metasurfaces .
The paper is organized as follows: in Sec. II we discuss the problem of nonlinear light scattering of a plane wave by a dielectric nanoparticle made of BaTiO or AlGaAs materials. Applying Green’s function approach, we calculate the efficiency of SHG and multipolar content of the second-harmonic (SH) field. We also consider the particular case of SHG through excitation of a single magnetic dipole mode. We discuss how the intensity and the far-field properties of the SH field vary while direction of the excited dipole moment changes relatively to the crystalline structure of material. By explicit calculations we show how the mode content of SH field varies. In Sec. III we derive selection rules which govern the channels of mode coupling at fundamental and SH wavelengths based on the symmetry of vector spherical harmonics and the crystalline structure. In Sec. IV we apply the formulated selection rules to explain the results obtained in Sec. II and build the complete table of possible generated multipoles for SHG process driven by dipole modes.
Ii Second-harmonic generation formalism
ii.1 Green’s function approach. Plane-wave excitation
We consider a spherical dielectric particle of the radius characterized by a frequency-dependent dielectric permittivity , embedded in a homogeneous host medium with . The nanoparticle is made of the material with a noncentrosymmetric crystalline structure, and its nonlinear electromagnetic properties are captured by the quadratic susceptibility tensor . While for AlGaAs the linear susceptibility tensor is isotropic, for BaTiO this tensor inherits the uniaxial crystal structure of the material. In this case, the anisotropy of SHG tensor and of the linear permittivity tensor are not two independent phenomena and have a common microscopic origin . However the anisotropy is rather weak in the case of BaTiO, it dramatically increases the complexity of the problem comparing to isotropic linear scattering. Hence, from now on we use the approximation of isotropic linear susceptibility tensor. The effect of anisotropy on the selection rules will be discussed in more details at the end of the paper in Sec. IV.
The problem of linear light scattering by a sphere is solved using the multipolar expansion following the Mie theory [30, 31]. In our work we consider time dependence of the fields in the form . For the illumination by the -polarized plane wave incident along the direction, the field inside the spherical nanoparticle () is expanded in vector spherical harmonics as follows:
where the wavenumbers , . Magnetic and electric spherical harmonics with the total angular momentum and the momentum projection , the indexes describing their parity with respect to the reflection along axis (or transformation), and the coefficients , are given in Appendix A, the superscript is used to define spherical Bessel functions.
The induced nonlinear polarization at the second-harmonic frequency is defined by the second-order polarizability tensor:
where are the Cartesian components of the fundamental field inside the nanoparticle. We take into account only the bulk nonlinear response leaving outside the consideration potential surface sources of SHG. The SH field outside the particle at can be found using dyadic Green’s function of a sphere:
satisfying the following equation where is the unit dyadic, for , and for .
The explicit multipole decomposition of the Green function is given in Ref. 32, and also in Appendix A. Substituting the expansion of the Green function in the form of the spherical waves into Eqs. (2),(3) we obtain the multipolar decomposition of the second harmonic field
Here, the denotation distinguishes between electric and magnetic harmonics, the superscript is used to define spherical Hankel functions of the first kind. The expansion coefficients are readily evaluated as a sum of overlap integrals between the two vector spherical harmonics at the frequency and one harmonic at the frequency, weighted by the tensor:
The explicit form of these coefficients is given in Appendix A.
Finally, the second-harmonic conversion efficiency , defined as the ratio of the total SH radiated power to the energy flux of the fundamental wave through the geometrical cross section of the particle, can be expressed through the coefficients as follows :
Using Eq. (6), we calculate the conversion efficiency for BaTiO and AlGaAs particles of subwavelength sizes under the plane-wave excitation. In this section, we assume that the main axes of crystalline lattice are oriented along the coordinate system: (see Fig. 1). In Section IV we will discuss other crystalline orientations. In the chosen coordinate system the second-order polarization of the BaTiO crystal has the following form:
where pm/V, pm/V, pm/V . In the principal axis system of the AlGaAs crystal, the tensor of the second-order nonlinear susceptibility contains only off-diagonal elements pm/V being non-zero if any of two indices do not coincide:
The fundamental wavelengths are fixed for BaTiO and AlGaAs to 1050 nm and 1550 nm respectively. These values were chosen in accordance with the typical experimental frequencies used for observation of SHG from these materials and correspond to the Yb laser (1050 nm)  or the Er doped fiber laser (1550 nm). Since AlGaAs has a higher refractive index () compared to BaTiO (), the particle sizes are within the same range.
The calculated dependences of SHG on the nanoparticle radius are shown in Figs. 2(a), and 3(a) demonstrating pronounced resonant structure. In order to distinguish between different multipolar resonances, we have separately calculated the contribution from each multipolar channel in Eq. (6) [see the colored curves in Fig. 2(a), and Fig. 3(a)]. The contributions of the harmonics with the same total angular momentum and different momentum projections are combined together. Identification of the harmonics with particular momentum projection contributing to the SH emission will be discussed in detail in Sec. IV (see Tables 3, 4, 5). We also support the SH spectra with the plot of the linear scattering spectra of a plane wave at the fundamental and SH wavelenghts in Figs. 2,3 b), clearly showing individual Mie resonances.
One can see that the peaks at the SH wavelength are modulated with the broad resonance at the fundamental wavelength, which is clearly seen by comparing the panels (a) and (b) in Fig. 2 and Fig. 3. As a result, the SHG efficiency increases by several orders of magnitude when approaching the magnetic dipole (MD) resonance at the fundamental wavelength [35, 36]. The dramatic enhancement is observed when the double-resonance condition is fulfilled [37, 38, 39] for instance at electrical octupole (EO) resonance [see Figs. 2, and 3 (a)]. Results of our analytical calculations are confirmed by full-wave numerical modeling performed with the finite-element solver COMSOL Multiphysics, following the procedure applied in Refs. [25, 9, 22, 21]. The multipolar amplitude coefficients are then numerically retrieved and reproduce Figs. 2,3.
The magnitude of the SH conversion efficiency is intensity-dependent , as it describes the two-photon process. Specifically, for a given intensity of the incident wave of GW/cm the conversion efficiency reaches the value of for BaTiO nanoparticle and for AlGaAs nanoparticles in the same radius range of around 200 nm. These values are about one order of magnitude higher than the experimental values measured for nanodisks in similar experimental conditions [12, 9]. This discrepancy can be related to the lower SHG efficiency from disk resonators studied in the experiments due to the substrate effects and the uncertainty of retrieving of the efficiency value from experimental data.
Another important feature is the particular multipolar content of the SH field. For instance one can notice that the MD is absent in the SH field generated in the BaTiO nanoparticle, and no electric dipole (ED) field is generated in the AlGaAs nanoparticle. This cancellation is dictated by the symmetry of the tensor and direction and polarization of the fundamental wave. It will be further illustrated in Sec. II.2, studied in detail from the symmetry point of view in Sec. III, and discussed in Sec. IV.
ii.2 Single-mode approximation
Here, we specifically focus on the SHG driven by the MD mode only. In the vicinity of pronounced resonances, the field distribution inside the particle excited by the fundamental wave can be approximated by the corresponding eigenmode [25, 21]. Selective and enhanced coupling to specific multipole modes can be facilitated by the beam engineering [40, 41]. If the refractive index is high enough, , the fundamental MD resonance dominates in the fundamental field in particular spectral region (around 220 nm radius for the fundamental wavelengths in Fig. 2, and Fig. 3). The case of SHG driven by MD excitation represents an instructive example for understanding the multipolar nature of the generated electromagnetic fields in Mie-resonant dielectric nanoparticles.
We employ a single-mode approximation and assume that the field inside the AlGaAs nanoparticle at is given by a MD mode profile with the -aligned magnetic moment:
This geometry corresponds to or , in Fig. 4. Integration of the trigonometric functions in Eq. (5) over the angles shows that within the framework of single-mode MD approximation (9) and crystalline axes of material oriented according to Fig. 1, the multipolar composition features electric octupole and magnetic quadrupole for the AlGaAs nanoparticle, allowing us to write down the expression for the field:
The multipolar amplitudes , and can be also conveniently found using the Lorentz lemma following the procedure described in Ref. , being alternative to the Green’s function integration in Section II.1. This allows us to write the expression for -amplitudes through transmission coefficient (see Ref. 21) of the incident spherical wave irradiating the particle. After some algebra, we obtain compact analytical expressions for the multipolar coefficients:
The relative contribution of different multipoles varies when the SH wavelength is tuned to corresponding Mie resonances. This immediately follows from the expressions for multipolar amplitudes Eq. (11). In Fig. 5 we trace this behaviour by plotting the dependence of the SH intensity on nanoparticle size. When the radius is increased, the contribution of EO mode starts to dominate over the magnetic quadrupole (MQ) changing the far-field radiation pattern from axially-symmetric for smaller particles to the multi-lobed pattern near the EO resonance at nm. The field distribution inside the nanoparticle (see insets in Figs. 5,6, 7) was obtained with the help of COMSOL Multiphysics package. The radiation patterns (see insets in Figs. 5,6, 7,8), showing the distribution of the generated SH intensity in the far-field, were plotted with the use of the formula Eq. (4) and were verified with COMSOL Multiphysics.
Rotation of the pump magnetic dipole in the plane ( in Fig. 4) enables the generation of the ED mode in AlGaAs nanoparticle, which is also illustrated in Table 1. The presence of the specific modes in the SH spectrum will be discussed in detail in Sec. IV basing on symmetry reasons. For the smaller nanoparticle of nm (Fig. 6(a)), being remote from the EO-resonant size, the EO contribution in the SH field is negligible. At or , the SH radiation is dominated by the magnetic quadrupole. If the pump magnetic dipole is rotated by , the ED relative contribution exceeds MQ and the radiation pattern significantly changes. For the larger nanoparticle of nm [Fig. 6(b)], the EO term dominates in the SH emission.
The results of analogous calculations for BaTiO particle are shown in Fig. 7. Here, two different orientations of the magnetic dipole ( and ) are shown and the resonant switching between dipolar, quadrupolar, and octupolar modes is observed.
In Fig. 8 we illustrate the effect of the MD rotation in the -plane () for the BaTiO particle of radius nm, corresponding to the MQ peak in Fig. 3(b). Rotation of MD in the -plane will not give any changes due to symmetry of BaTiO lattice with respect to this rotation. At the induced nonlinear source does not contain a MQ component, and, thus, weak SHG is determined by the non-resonant ED. When the angle is increased, the total SHG intensity grows and the leading contribution to the SH radiation originates from the resonant multipole MQ.
Iii Symmetry analysis for the second-harmonic generation
In this section, we analyze the possibility of SHG through different multipole channels which is determined by the particular symmetry of modes at the fundamental and SH frequencies. The integrals over the nanoparticle volume
determine the contributions of the multipoles , to the SHG expansion coefficients in Eq. (5). Here, the indices ,, stand for the parity and the projection of vector spherical harmonics . For each particular value of indices such integrals can be readily calculated analytically, and a large number of them turn out to be zero. Our goal is to reveal the general origin of these cancellations. All our considerations are based on the following general theorem [42, 43]. Let be one of the basis functions of an irreducible (non-unit) representation of a system’s symmetry group. Then the integral of this function over the configuration space of the physical system vanishes identically: . In order for the integral to be non-zero, the integrand must contain a term that is invariant when any of the symmetry operations of the group are applied, otherwise the integral vanishes.
In the considered case, the cancellations stem from both (i) the microscopic crystalline symmetry of the material and (ii) from the macroscopic spherical symmetry of the nanoparticle as a whole. In order to illustrate this distinction, we show in Fig. 9 (a) a tetrahedral nanoparticle cut from a material with point group symmetry, e.g. AlGaAs, and in Fig. 9 (b) the spherical nanoparticle made of the crystal with point group symmetry. Then, since tetrahedron has the same symmetry, all the allowed transitions will be defined by the crystalline symmetry only. However, spherical symmetry of the nanoparticle in Fig. 9 (b) imposes additional restrictions on the second harmonic generation, absent in a tetrahedron. The restrictions are provided by a certain parity of sphere eigenmodes with respect to the symmetry transformations of a sphere. Hence, we can make use of selection rules based on spatial parity of eigen modes. As such, most of the cancellations of the integrals Eq. (12) stem from simple spherical symmetry considerations, discussed in the following Sec. III.1. The rest of the relevant cancellations can be explained as inherited from the crystal point group symmetry and will be considered in Sec. III.2.
We note, that in our consideration we neglect the roughness of the spherical particle surface imposed by the crystalline structure as we assume nanoparticle containing large enough number of atoms. The possible nonlocal corrections to the linear dielectric response of the nanoparticle, arising from the spatial dispersion of the permittivity, and sensitive to the difference between tetradehral and spherical symmetry , are also neglected here, and the particle is described by the local isotropic permittivity tensor.
iii.1 Restrictions imposed by the spherical symmetry
The symmetry analysis of the integrals Eq. (12) becomes more straightforward when the spherical harmonics are presented in a vector form. To this end we use the following relationship between the Cartesian basis vectors and the electric dipole harmonics in the limit : , . This allows us to rewrite the integral Eq. (12) as
Let us consider the integral (13) in more detail. First of all, it contains a sum of several integrals of three scalar products of vector spherical harmonics, where each term corresponds to one of -tensor components. We are seeking for the cases when the integrand is invariant under the transformations of the symmetry group. We expect the integral to be non-zero, if at least one of the integrand terms contains a function invariant under all rotations and inversion transformation. The scalar products entering Eq. (13) can be readily expanded over the scalar spherical harmonics, see Refs. [45, 46] and Appendix B. Afterwards, the resulting integrals can be analyzed for different -tensor components and three general Rules , governing whether the integrals are zero or not, can be formulated for each component separately.
Rule A: Parity under inversion and reflection in the plane.
The vector spherical harmonics (Appendix A) are transformed in the same way as the real scalar spherical harmonics [47, 48, 49, 50] under the coordinate , while under and acquire a sign and acquires a sign , so the parity of magnetic vector harmonics is inverse to the parity of electric and scalar harmonics. We introduce the parity indices for , and for describing the behavior of spherical harmonics under the spatial inversion. Another parity index describes the behavior of the harmonics under the reflection in the plane, equivalent to the change of the azimuthal angle . The functions , and are even with respect to such reflection, , while the functions and are odd, . Thus, the inversion and reflection parity selection rules can be summarized as
The rule Eq. (14) is applied to the right-hand side of Eq. (13) as a whole. The rule Eq. (15) is applied to the individual products of different Cartesian components corresponding to each nonzero element of the nonlinear susceptibility tensor in Eq. (13). The factor in Eq. (15) is the parity of the product under the reflection, which is illustrated in Fig. 10. In the following, we will also use notation of or for the parity indices corresponding to the fundamental or the SH modes.
Rule B: Conservation of the angular momentum projection.
Once the scalar products in Eq. (13) are calculated, the matrix element is reduced to the overlap integral of scalar spherical harmonics. The angular momentum projection rule for the tesseral harmonics can be written as
Rule C: Triangle inequality for the total angular momentum.
This rule can be formulated as
where the index denotes the number of electric harmonics under the integral (12). The presence of the index stems from the fact that the Cartesian components of vector electric harmonics with the total angular momentum include only the scalar harmonics with the momentum , while the projections of magnetic harmonics include only the states with the same angular momentum .
Now, let us illustrate the rules obtained above with some specific examples. We start with examining the possibility of the generation of -polarized electric dipolar mode in a AlGaAs nanoparticle by combining the -polarized magnetic dipolar mode and the -polarized electric dipolar mode . Here , so such generation is prohibited by the first selection rule. Next, we try to replace by some electric quadrupolar mode, for example, . For this mode we have , according to the Fig. 10, and . Hence, such process is not prohibited by the first two rules. But the sum of angular momentum projections (16) for considered three multipoles is always odd, while for the tensor component it is even (Fig. 10). This means that the total sum is never zero, and such generation process is prohibited by the second rule.
These rules above provide large number of possible cancellations, however, some exceptions are possible due to the properties of scalar products. To get all cancellations for the specific tensor component, we either should use the rules of how three scalar products are coupled, following the algorithm given in Appendix B, or apply additional symmetry reasons, discussed below.
iii.2 Restrictions imposed by the crystal point group symmetry
In the previous Sec. III.1, we have separately considered the cancellations of the terms in Eq. (12) corresponding to the individual components of the -tensor. However, some of the components are equal due to the crystal point group symmetry, that can result in additional cancellations after the summation over tensor components is performed. Such mutual cancellations are taken care of by the theory of representations of symmetry groups in a universal automatic fashion. The detailed analysis is given below.
Transformation of the matrix elements under symmetry operations
In order to determine the behavior of the matrix elements Eq. (12) under the application of the point symmetry group operation, we consider how vector spherical harmonics are transformed. Here, one has to take into account that the transformed harmonic is in general expressed via a sum of the harmonics with different angular momentum projections and parity , but the same polarization (M or N) and the same total angular momentum [47, 48, 49, 50]. The scalar products, entering the integral Eq. (13) are transformed under the symmetry operation as
Here are the representation matrices of the symmetry operation with momentum for tesseral harmonics , in case of rotations they reduce to the combinations of the Wigner matrices. As an example we consider the case when the harmonics are the electric dipole harmonics, , and is a rotation around the axis, e.g. , , . In this case Eq. (18) simplifies to
i.e. a scalar product of two electric dipole modes is transformed as a second-rank cartesian tensor.
Application to harmonic generation
The most general consideration would require an expansion of the tensors Eq. (7), Eq. (8) of BaTiO and AlGaAs, transforming under the spherical symmetry operations according to Eq. (21), over the irreducible representations of the symmetry group. However, in practice it turns out that the relevant cancellations of the matrix elements Eq. (13), not captured in the previous Sec. III.1, can be explained in a simpler way. Instead of the whole group it suffices to apply a crystal point subgroup of the group, i.e. to use a smaller set of symmetry operations. When the crystal point group operation is applied, the tensor stays invariant, which means that , and . Hence, the matrix elements in Eq. (20) are transformed as a direct product of the three representations governing the transformation of the corresponding vector spherical harmonics. In order to stay non-zero, the integrals Eq. (20) should contain a combination, invariant to the symmetry operation of the crystal. More formally, the reducible representation governing the transformation Eq. (13) should contain an identity representation.
The symmetry analysis of the second harmonic generation is then reduced to (i) expanding the vector spherical harmonics over the basis functions of the irreducible representation of the crystal point symmetry group and (ii) using the Clebsh-Gordan coefficients available for all point groups [52, 43]. The classification of the dipole, quadrupole and octupole spherical harmonics for and point groups of AlGaAs and BaTiO, respectively, is given in Appendix C and Table 6. The examples of basis functions of irreducible representations, which behave in the same way under symmetry transformations as the spherical harmonics, are also given. The expansion has been done using the transformation properties of vector spherical harmonics. For example, we see from Table 6 that the electric dipole modes are transformed in the group according to the irreducible representation, i.e. as components of the radius-vector . Conversely, the magnetic dipole modes behave as pseudo-vector components, i.e. according to the representation.
Such approach allows us to find all the selection rules of the nanoparticle with the same or higher symmetry as the material (Fig. 9a)). It can be a tetrahedral nanoparticle for or a quadrangular pyramid nanoparticle for . This is possible due to the fact that we consider the integrand behavior under such transformations only. The further cancellations can appear due to the scalar product properties, for example, -component of harmonic is zero, so it can provide some extra restrictions (see Appendix B ).
Let us apply the developed selection rules to the cases studied in Sec. II, where we already discussed the absence of particular harmonics in the generated field.
Plane-wave excitation. Under the excitation of BaTiO (AlGaAs) nanoparticle with a plane wave, we observed the absence MD (ED) modes in the SH field. It is illustrative to start with the restrictions imposed by the spherical symmetry of modes that account for the most of the selection rules.
Here, we will consider only the dipole terms in the excitation, and higher modes can be treated analogously. Applying Rule A to the BaTiO nanoparticle in the case of dipole modes () we do not obtain any restrictions due to the inversion parity, because we both terms and are contained in the fundamental field. One can find that the reflection parity of the SH mode should be . Indeed, according to the Mie theory generated dipole modes at the fundamental frequency can be only and , and for both of them , while for BaTiO tensor (see Fig. 10). From the angular momentum projection conservation Rule B, we find the limits for angular momentum of SH modes. For the tensor components the sum is even according to Fig. 10. Thus, should also be even, and from the Mie theory it follows that , which makes to be even. This immediately rules out all magnetic dipole modes as the only dipole mode with even and is , which is identical to zero. Electric dipole mode has the same reflection parity and, thus, is allowed in the SHG process (see Table 3). These reasons also show that the higher order magnetic and electric modes can also exist. According to Rule C the highest possible harmonic generated from the dipole modes is the electric mode with as shown in Table 3: .
The selection rules for AlGaAs are shown in the Table 5. The same arguments as in the case of BaTiO can be applied, while considering dipole channels of SHG. The only difference is that the parity of tensor components for AlGaAs (see Fig. 10). Rule A will be fulfilled if the parity of one of the modes will be changed in sign, allowing transition. Another possible channels or are forbidden, as it would require generation of mode, which is identical to zero. Moreover, the dipole modes generation in SH field will be still prohibited even if the higher order modes will be excited at fundamental wavelength.
Single magnetic mode excitation. In Sec. II.2, we have discussed the excitation of the SH mode with a single magnetic mode at the fundamental frequency. Let us first study in more detail the case of -oriented dipole . Applying derived selection rules, one can get from Rule C that the highest possible generated mode is the octupole mode . For BaTiO nanoparticle from Rule B, we have already established that should be even. The inversion and reflection parities from Rule A gives us that and . For total angular momentum value of this means that only electric modes should be generated (inversion rule), and they should be even (reflection rule), which gives us for even only two possible modes: and . For only odd magnetic quadrupole mode possesses required reflection and inversion parity, which gives us only contribution as . These selection rules are summarized in the highlighted row of Table 2, which also corresponds to highlighted region of the extended Table 3.
Until now we have considered only one orientation of the crystalline lattice, shown in Fig. 1. However, the different orientation of the BaTiO lattice provides other selection rules. For instance, one can rotate the BaTiO lattice by 90 obtaining , which changes the parity indices , and should be odd. The resulting selection rules are summarized in Table 4. One can see that harmonics with different projection can be generated, allowing both MD and ED channels. This provides us an opportunity to control the SH signal by changing the relative orientation between the field polarization and the crystalline lattice.
The same approach allows us to analyze the modes generated at SH by pumping at single MD mode in AlGaAs nanoparticle, as shown in Table 1. The shaded region describes excitation of the MD mode corresponding to -oriented magnetic dipole (). The same argument as in the case of BaTiO allows up to octupole mode generation. We have already discussed in this section that the generation of ED mode is possible in AlGaAs due to the parity. Because of that, the inversion and reflection parity values should be equal to and . Thus, for is the only nonzero mode satisfying the parity conditions, while for the even magnetic modes have proper parity, thus, and are present (shaded rows in Tables 1 and 5).
The selection rules provided by the crystalline lattice symmetry. So far we have discussed the selection rules which were governed by the symmetry of the vector spherical harmonics and related conditions A, B, and C. However, there are SHG channels, which are allowed by the mode symmetry, but become restricted due to crystalline symmetry only.
For the SHG by rotated MD in AlGaAs nanoparticle for (see Table 1) modes and are present in the fundamental field, while the channels of generation of higher quadrupole modes are forbidden , however some of them satisfy the mode symmetry rules. These processes are restricted by the crystalline symmetry rules discussed in Sec. III.2: in the symmetry group the magnetic dipole modes are transformed as components of the pseudo-vector ( representation).The modes , , also behave under the symmetry operations as components of a pseudo-vector ( representation), see the Table 6. Now, the physical question, of whether the SHG process is possible, is reduced to the mathematical question of whether the direct product contains . The 9-dimensional reducible representation is equal to a direct sum of irreducible representations [52, 43]. If we label the 3-pseudovectors, forming the basis of the representation as and , the nine linear combinations transforming to , , , are (), and (), and two cyclic permutations () and (). The latter must be understood componentwise. We are interested only in the magnetic dipole contribution, i.e. in the pseudovector transforming according to . However, in our case and are equal, since both modes at the first harmonic frequency belong to the same electric field. As such, the vector product is zero and the conversion from the magnetic dipole modes to the ,, quadrupolar modes is forbidden by the crystal symmetry. For the similar reasons, such process is prohibited for the conversion from the electric dipole modes and . On the other hand, a sum frequency generation process, when the incident modes have different frequency, can be possible, since in general is not zero.
Anisotropy of the linear response of the nanoparticle. Finally, let us briefly discuss the effect of the uniaxial symmetry of the linear response of the dielectric tensor, present for BaTiO on the obtained selection rules for the second harmonic generation. The impact of anistotropy on linear scattering has been studied in details in Ref. 53, 54. However, for the considered range of parameters the anisotropy is not very strong, namely and at the fundamental wavelength of 1050 nm  and and at the SH wavelength of 525 nm , a rigorous extension of the approach presented in this paper to the anisotropic case manifests itself a complicated problem. Thus, we have appleid numerical simulation method in order to check the effects of the present material anisotropy on the SH field multipole content. The spectral dependence of the SHG cross section is shown in Fig. 11 similarly to Fig. 2 but with account for anisotropic permittivity tensor. Simulations results have shown that the difference in the nonlinear response is rather weak, and the multipolar contents is generally preserved for the given set of the parameters.
The further speculations on the influence of the anisotropy will bring us to the conclusion on the selection rules modification. In general, the linear scattering of a plane wave on an isotropic particle preserves both the multipole order , and the electric or magnetic parity, namely each spherical harmonic contained in a plane wave generates a scattered harmonic with the same numbers , and the same electric/magnetic parity. In the process of anisotropic particle scattering the mode numbers are not conserved [54, 57] and multipole orders , their projections (in case of arbitrary orientation of the optic axis of the crystal), and electric and magnetic degrees of freedom are getting mixed. The SHG process will also entangle the multipole orders due to the structure of the anisotropic Green’s function. These two processes change the selection rules A-C. However, the formulation of the exact selection rules in this case is a matter of the future work, there will be a particular resemblance to the case of the SHG from a cylindrical nanoparticle with isotropic linear permittivity tensor .
In conclusion, we have theoretically analyzed the second harmonic generation by spherical dielectric nanoparticles made of materials with non-zero bulk second order nonlinear susceptibility tensor . Considering two typical crystalline solids, BaTiO and AlGaAs, we have studied the intensity of SHG under a plane wave illumination and analyzed the contribution of different multipole components into the total SH intensity. We have shown that under the resonant excitation of a single magnetic dipole mode one can achieve control of the directionality of SH emission by rotating the dipole moment with respect to the material’s crystalline lattice. Finally, we have developed a symmetry approach which provides an explanation why only particular modes can be observed in the SH field and defined general selection rules for SHG. Our symmetry analysis fully agrees with numerical and analytical results, and also demonstrates promising predictive power, which can be used for design of efficient nonlinear light sources based on nanoparticle ensembles.
Acknowledgements.The authors acknowledge useful discussions with S.E. Derkachev, A. A. Nikolaeva, M.O. Nestoklon, E.L. Ivchenko, A.I. Smirnov. The work was supported by the Russian Foundation for Basic Research (Grant No. 18-02-00381, 18-02-01206). Numerical modeling, performed by D.S., was supported by the Russian Science Foundation (Grant No. 17-12-01574). A.P. and M.P. have been supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “Basis”.
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Appendix A Vector spherical harmonics
Vector spherical harmonics used above are defined as
are the scalar spherical functions, proportional to the tesseral spherical functions. Functions can be replaced by spherical Bessel functions of any type, is .
The Mie coefficients for the field inside the nanoparticle can be expressed as follows:
here is , is