Exploring optical resonances of nanoparticles excited by optical Skyrmion lattices

Qiang Zhang; Zhenzhen Liu; Feifei Qin; Shang Jie Zeng; Dasen Zhang; Zhiyuan Gu; Xiangli Liu; Jun-Jun Xiao

doi:10.1364/OE.27.007009

1. Introduction

Exploring optical resonances of nanoparticles excited by light is one of the fundamental topics in nanophotonics. For metallic nanoparticles made by noble metals (gold, silver, etc.), the prominent optical resonances are known as localized surface plasmons coming from the coherent oscillation of free electrons driven by light [1]. Besides, optical resonances are also sustained by high-refractive-index dielectric nanoparticles, referring as the cavity-like Mie resonances [2–5]. The enhanced light-matter interactions by these resonances usually lead to strong light scattering or/and absorption, and significant confinement and enhancement of near-fields in sub-wavelength scale [6–8]. Moreover, the resonant responses of such nanoparticles can be engineered by adjusting their size, material components and ambient environment [9,10]. Because of these attractive optical properties, resonant nanoparticles have a wide spectrum of applications including, but not limited to, optical nanoantennas [11–13], surface enhanced Raman scattering [14–16], biosensing and imaging [17,18], structural color printing [19,20], spontaneous emission enhancement [21,22], optical trapping [23,24], and nonlinear harmonic generation [25,26].

Flexible manipulation of optical resonances in nanoparticles, such as excitation and spectral tuning, is crucial for the aforementioned applications. The excitation efficiency of a specific optical resonance depends on both the spectral and polarization matching between the impinging light and the resonant mode. On the one hand, optical resonances in nanoparticles have been widely studied by conventional optical spectroscopies using light sources with spatially homogenous polarization states, such as linearly, elliptically or circularly polarized plane waves. On the other hand, structured light fields with spatially inhomogeneous polarization states have received much attention recently in terms of exploring some intriguing optical resonances that are usually difficult to be excited by traditional waves. In this respect, tightly focused radially and azimuthally polarized cylindrical vector beams have shown their capabilities of selective switching of individual multipole resonances in nanoparticles [27–29] and enabling efficient excitation of optical dark modes [30–33].

Very recently, Tsesses et al. report a type of optical fields with topological stability called optical Skyrmion lattices (OSLs) in evanescent electromagnetic fields [34]. Similar to the magnetization vectors of Skyrmion lattices in chiral magnets [35–37], OSLs possess three-dimensional polarization vectors in a two-dimensional space, giving an integer Skyrmion number. As such, the intrinsic topological property and exotic field configuration promise OSLs many potential applications in optics and photonics, such as optical information processing and light emission control. Particularly, one may realize that OSL is actually a kind of structured near-field with spatially varied polarization vectors. Thus it shall be interesting to explore the interaction between nanoparticles and an OSL. Here we present such a study by investigating the scattering responses of nanoparticles placed in an OSL. We firstly construct OSLs at the interface between a dielectric substrate and air by utilizing multiple evanescent waves through total internal reflection. Then the multipole scattering responses of several representative nanoparticles placed in the OSLs are numerically studied. It is found that OSLs have polarization properties resembling those of tightly focused radially and azimuthally vector beams, providing an excellent platform to inspect various kinds of optical resonances in nanoparticles. The results indicate that as a kind of structured near-field, OSLs may be applied in nonlinear microscopy [38,39], active tuning of unidirectional scattering of nanoantennas [40,41], optical trapping [42], etc.

2. Results and discussion

2.1 Generation of OSLs based on total internal reflection

As sketched in Fig. 1(a), an OSL can be generated at the interface between a dielectric medium (refractive $n_{1} > 1$ ) and the air (refractive index $n_{2} = 1$ ) by interfering three pairs of counter-propagating evanescent waves (red arrows labeled with w₁~w₆) that are obtained through total internal reflection. The propagating directions of the adjacent pairs are different by an angle of $π / 3$ [34]. Taking the pair containing waves w₁ and w₂ for example, the principle of total internal reflection is shown in Fig. 1(b). Two incident plane waves $w_{1}^{i}$ and $w_{2}^{i}$ (black arrows) come from the dielectric side with the same incidence angle $θ_{i}$ and opposite in-plane wave vectors ( + x for $w_{1}^{i}$ and –x for $w_{2}^{i}$ ). Total internal reflection happens when the incidence angle is larger than the critical angle $θ_{i} > θ_{c}$ , yielding two counter-propagating evanescent waves w₁ and w₂ (red arrows) over the interface. The other two pairs of evanescent waves are generated in the same way, all with the same incidence angle and field amplitude but in opposite in-plane propagating directions.

Fig. 1 (a) Sketch of generating an OSL at the interface between a dielectric medium (refractive $n_{1} > 1$ ) and the air (refractive index $n_{2} = 1$ ) by interfering three pairs of counter-propagating evanescent waves (red arrows labeled w₁~w₆). (b) The pair of counter-propagating evanescent waves w₁ and w₂ (red arrows) is generated by two incident waves $w_{1}^{i}$ and $w_{2}^{i}$ (black arrows) coming from the dielectric side with the same incident angle and opposite in-plane propagating directions. The blue arrows represent the reflected waves $w_{1}^{r}$ and $w_{2}^{r}$ . (c) Vector decomposition of the in-plane ( $x y$ plane) magnetic field components ( $H_{i x}^{ϕ}$ and $H_{i y}^{ϕ}$ ) of a TM polarized incident wave ( $H_{i}^{ϕ}$ ). The in-plane propagating direction of the incident wave is determined by the azimuthal angle $ϕ$ between the in-plane wave vector $k_{∥}$ and the $+ x$ axis.

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The generated OSL can be either electric-type or magnetic-type, depending on the polarization sates of the incident waves, i.e. TM for electric-type and TE for magnetic-type. Here we take the generation of an electric-type OSL for the illustration. The in-plane magnetic components of the TM polarized incident waves are $H_{i}^{ϕ} = H_{i x}^{ϕ} \hat{x} + H_{i y}^{ϕ} \hat{y}$ , where the superscript $ϕ$ indicates the azimuthal angle which is defined as the angle between the in-plane wave vector $k_{∥}$ and $+ x$ axis as shown in Fig. 1(c). Based on the coordinate and vector decompositions illustrated in Fig. 1(c), the magnetic field components of the incident wave infrequency domain are readily written as:

(\begin{matrix} H_{i x}^{ϕ} (ω) \\ H_{i y}^{ϕ} (ω) \end{matrix}) = (\begin{matrix} - \sin (ϕ) e^{- j k_{z 1} z} H_{0} e^{- j (k_{x}^{ϕ} x + k_{y}^{ϕ} y)} \\ \cos (ϕ) e^{- j k_{z 1} z} H_{0} e^{- j (k_{x}^{ϕ} x + k_{y}^{ϕ} y)} \end{matrix}),

where

ω

is the angular frequency and

H_{0}

is a real constant denoting the field amplitude. The in-plane and out-of-plane wave vectors of the incident wave are given by

k_{x}^{ϕ} = k_{∥} c o s (ϕ)

,

k_{y}^{ϕ} = k_{∥} s i n (ϕ)

, and

k_{z 1}^{2} + k_{∥}^{2} = n_{1} k_{0}^{2}

, where

k_{0}

is the wave vector in vacuum. The transmission Fresnel coefficient for TM-polarized wave in terms of magnetic fields is given by

t_{p} = 2 n_{2}^{2} k_{z 1} / (n_{2}^{2} k_{z 1} + n_{1}^{2} k_{z 2})

[43], where the wave vector components in air satisfy

k_{z 2}^{2} + k_{∥}^{2} = n_{2} k_{0}^{2}

. Then, the transmitted magnetic field

H_{t}^{ϕ}

is directly obtained as

H_{t}^{ϕ} = t_{p} H_{i}^{ϕ}

. Based on the Maxwell equations, the transmitted electric field can be obtained as

E_{t}^{ϕ} (ω) = \nabla \times H_{t}^{ϕ} (ω) / j ω n_{2}^{2} ε_{0}

, where

ε_{0}

is the permittivity of the vacuum and

j = \sqrt{- 1}

. When total internal reflection happens, the transmitted wave becomes evanescent and the z-component wave vector

k_{z 2}

is pure imaginary. The electric field of the generated electric-type OSL is simply a summation of three pairs of counter-propagating transmitted waves with

ϕ = - π / 3, 0

and

π / 3

:

(\begin{matrix} E_{x} (ω) \\ E_{y} (ω) \\ E_{z} (ω) \end{matrix}) = E_{t} e^{- | k_{z 2} | z} (\begin{matrix} \frac{| k_{z 2} |}{k_{∥}} \sum_{ϕ = - \frac{π}{3}, 0, \frac{π}{3}} \cos (ϕ) \sin (k_{∥} [\cos (ϕ) x + \sin (ϕ) y]) \\ \frac{| k_{z 2} |}{k_{∥}} \sum_{ϕ = - \frac{π}{3}, 0, \frac{π}{3}} \sin (ϕ) \sin (k_{∥} [\cos (ϕ) x + \sin (ϕ) y]) \\ \sum_{ϕ = - \frac{π}{3}, 0, \frac{π}{3}} \cos (k_{∥} [\cos (ϕ) x + \sin (ϕ) y]) \end{matrix}),

where

E_{t} = - 2 k_{∥} H_{0} | t_{p} | / ω ε_{0}

. Note that here we omit the global phase caused by the complex transmission coefficient

t_{p}

to get real electric field components in Eq. (2). The OSL has a hexagonal lattice structure and the lattice size

p_{s k y}

can be obtained from Eq. (2) as

p_{sky} = 2 λ_{0} / (α \sqrt{3})

with

α = k_{∥} / k_{0}

.

2.2 Field configuration of electric-type OSL

Without loss of generality, we consider an electric-type OSL generated at the glass-air interface ( $n_{1} = 1.5$ ) with incident angle $θ_{i} = 1.1 θ_{c}$ at wavelength $λ_{0} = 600$ nm. Figure 2(a) shows the vector representation of the unit electric field vector $\hat{e} = E / | E |$ which measures the local electric field direction (polarization vector) of the lattice. It is seen that the OSL has spatially varied polarization vectors which possess an integer topological invariant, namely the Skyrmion number [34]:

N_{s k} = \frac{1}{4 π} \int_{A} \hat{e} \cdot (\frac{\partial \hat{e}}{\partial x} \times \frac{\partial \hat{e}}{\partial y}) d a,

where the integration area

A

covers the whole unit cell.

Fig. 2 Field configuration of an electric-type OSL generated through total internal reflection at the glass-air interface with $θ_{i} = 1.1 θ_{c}$ at wavelength $λ_{0} = 600$ nm. (a) Vector representation of the unit electric field vector $\hat{e}$ at the $x y$ plane. (b) Variation of the electric field along the lattice vector represented by the green arrow in lower-left inset. The left inset shows the amplitude distribution of the out-of plane electric field ( $| E_{z} |$ ) in one unit cell and the right one shows that of the in-plane electric field $| E_{∥} |$ with the field vectors represented by the white arrows. (c) Vector representation of the unit magnetic field vector $\hat{h}$ at the $x y$ plane. (d) Magnetic field components in one unit cell along the lattice vector. Inset shows the amplitude distribution of the in-plane magnetic field $| H_{∥} | = | H_{x} \hat{x} + H_{y} \hat{y} |$ and field vectors (white arrows) of the magnetic field in one unit cell.

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To further understand the unique field configuration of the OSL, the electric field distributions in one hexagonal unit cell are examined in detail in Fig. 2(b). The curves in Fig. 2(b) represent the three electric field components as a function of the position (denoted as $s$ ) along one of the lattice vectors (represented by the green arrow in the left inset) in one unit cell ( $- 0.5 p_{sky}$ to $0.5 p_{sky}$ ). It is seen that the in-plane electric field components ( $E_{x}$ and $E_{y}$ ) vanish at the center of the unit cell where the out-of-plane component ( $E_{z}$ ) maximizes. This can be more clearly seen in the amplitude distributions of $| E_{z} |$ and $| E_{∥} | = | E_{x} \hat{x} + E_{y} \hat{y} |$ plotted in the left and right insets, respectively. It is also noticed that the symmetry along the lattice vector with respect to the center is opposite for the in-plane (odd symmetry) and out-of-plane (even symmetry) electric field components. More importantly, the in-plane electric field vectors close to the center of the unit cell (see the white arrows in the right inset) are almost radially polarized. But in the marginal regime of the unit cell, the azimuthal field component cannot be ignored. Accordingly, the in-plane magnetic field components in each unit cell of the OSL form a vortex-like distribution as shown by the vector representation of $\hat{h} = H / | H |$ in Fig. 2(c). Similar to the in-plane electric field components, the in-plane magnetic field vanishes at the center of the unit cell and is also odd symmetry with respect to the center as shown in Fig. 2(d).

2.3 Selective excitation of electromagnetic multipoles by OSLs

In view of the unique field configuration of the OSL shown in Fig. 2, we next focus on investigating the scattering properties of nanoparticles placed in an OSL. The total and multipole decomposed scattering cross sections (SCSs) of the nanoparticles are numerically calculated (see Appendix A). Firstly, we choose a Si (refractive index 3.5) nanosphere with diameter of 200 nm which supports both electric and magnetic multipole Mie resonances. Figure 3(a) shows the total and multipole decomposed SCSs of the Si nanosphere free standing in air excited by a linearly polarized plane wave (see the inset). The multipole decomposition is calculated up to quadrupole order including electric dipole (ED), magnetic dipole (MD), toroidal dipole (TD), total electric dipole (TED), electric quadrupole (EQ) and magnetic quadrupole (MQ). Here TED is the coherent superposition of ED and TD with dipole moment $P + i k_{0} T$ , where $P$ and $T$ are the electric and toroidal dipole moment, respectively [44–46]. All the considered multipole moments are defined in Cartesian coordinate system and determined by the polarization current induced in the nanoparticles (see Appendix B). It shows that the predominated multipoles of the silicon nanosphere excited by the plane wave in the wavelength range 500 nm~900 nm are ED (red line), MD (blue line) and MQ (wine line). In the short wavelength range, the contribution from the TD is observable but still weak.

Fig. 3 (a) Total and multipole decomposed SCS spectra of the Si nanosphere excited by a linearly polarized plane wave. Inset shows the excitation configuration. (b) Total SCS of the Si nanosphere excited by an electric-type OSL as a function of the wavelength and particle position along the lattice vector represented by the red arrow in the inset. (c) and (d) show the total and multipole decomposed SCS spectra of the Si nanosphere placed at the center ( $s = 0$ ) and off-center ( $s = 0.3 p_{sky}$ ) of the unit cell, respectively. Insets show the positions (red spots) of the particle center in the hexagonal unit cell.

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Then the Si nanosphere is placed atop the glass surface where an OSL is generated through total internal reflection with $θ_{i} = 1.1 θ_{c}$ . Figure 3(b) shows the total SCS of the Si nanosphere excited by the OSL as a function of the wavelength and particle position (denoted by $s$ ). Considering the field symmetry, the particle is moved from the center ( $s = 0$ ) to the margin ( $s = 0.5 p_{sky}$ ) of the unit cell along the lattice vector represented by the red arrow in the inset (same as the one discussed in Fig. 2). Clearly, the scattering of the Si nanosphere strongly depends on its position inside the OSL. Moreover, the scattering intensities in different wavelength ranges have distinct dependences on the particle position. To study this feature in more detail, the total and multipole decomposed SCS spectra of the particle placed at the center of the unit cell ( $s = 0$ ) are given in Fig. 3(c). We note that here the multiple decomposed SCS is only attributed to the scattered wave directly by the particle in the air region. A more rigorous multipole decomposition needs to consider the contributions from the reflected and transmitted waves when a substrate is present, yet requiring much more tedious math [47,48]. Nevertheless, ignoring these contributions will not influence the identification of the multipoles in the nanparticles due to the small refractive index of the glass substrate. Compared to the results shown in Fig. 3(a), it is found that the magnetic multipoles (MD and MQ) are killed when the nanosphere is located exactly at the center of the unit cell, leaving pure electric multipoles (ED and TD). As the particle moves away from the center, the magnetic multipoles show up and even become dominated in a certain position range. For example, Fig. 3(d) shows the results of the Si nanosphere placed at $s = 0.3 p_{sky}$ (see the red spot in the inset). Two scattering peaks at 730 nm and 505 nm can be attributed to the MD mode and MQ mode, respectively, while the ED mode strongly mitigates.

More detailed multipole decomposed SCSs of the ED, TD, MD, and MQ for the Si nanosphere as a function of the wavelength and particle position in the lattice are given in Figs. 4(a)-(d) in sequence. It is seen that the electric and magnetic multipoles are efficiently excited when the particle is placed at relatively separated regimes in the lattice. This is essentially related to the unique field configuration of the OSL as discussed in Fig. 2, which makes it possible to selectively excite electric and magnetic modes by placing a nanoparticle at proper positions in an OSL.

Fig. 4 Multipole decomposed SCS of the Si nanosphere excited by the electric-type OSL as a function of the wavelength and particle position. (a) The SCS of the ED as a function of the wavelength and particle position, (b) the SCS of the TD as a function of the wavelength and particle position, (c) the SCS of the MD as a function of the wavelength and particle position, and (d) the SCS of the MQ as a function of the wavelength and particle position.

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2.4 Efficient excitation of TD and plasmonic dark modes by electric-type OSL

Besides the spatially varied field intensity, the polarization character of the OSL is another important factor that affects its interaction with a nanoparticle. We particularly notice that magnetic field of an electric-type OSL has a vortex-like pattern [see Figs. 2(c) and 2(d)], reminiscent of a TD mode [44,49,50]. Thus, it is reasonable to expect that the OSL is favorable for the excitation of a TD mode. For a Si nanosphere, TD response can be seen in short wavelength range but with quite weak strength as shown in Figs. 3(a) and 3(c). Instead, a much stronger TD response can be obtained in a flat Si nanoparticle (for example a Si nanodisk). Figure 5(a) shows the total and multipole decomposed SCS spectra of a Si nanodisk excited by a plane wave polarized normal to the disk surface (see the excitation configuration in the inset). The diameter and thickness of the disk is 300 nm and 100 nm, respectively. The SCS of the TD shown in Fig. 5(a) (olive line) confirms a significant toroidal response for wavelength below 600 nm, which is, however, mixed with other multipoles including ED, MD and MQ. Similar to the case of Si nanosphere, the scattering of the Si nanodisk placed in an OSL strongly depends on its position in the lattice, as seen in the total SCS spectral map given by Fig. 5(b). When putting the Si nannodisk at the center of the unit cell, the magnetic multipoles of the Si nanodisk vanish and only the ED and TD remain, as seen in Fig. 5(c). Particularly, a pure TD mode is obtained at wavelength 575 nm whereED approaches to zero. Such pure TD mode is characterized by the strongly confined local magnetic field inside the nanodisk in a vortex pattern as shown in the inset of Fig. 5(c). Moving the Si nanodisk away from the center devastates the symmetry of the magnetic field with respect to the particle center, resulting in the expected excitation of magnetic mulitipoles (MD and MQ). Figure 5(d) shows the SCS spectra of the Si nanodisk placed at $s = 0.3 p_{sky}$ , from which we can see that the TD response is negligible in the whole wavelength band.

Fig. 5 (a) Total and multipole decomposed SCS spectra of the Si nanodisk excited by a linearly polarize plane wave. Inset shows the excitation configuration. (b) Total SCS of the Si nanodisk excited by an electric-type OSL as a function of the wavelength and particle position along the lattice vector represented by the red arrow in the inset. (c) and (d) show the total and multipole decomposed SCS spectra of the Si nanodisk when it is placed at the center ( $s = 0$ ) and off-center ( $s = 0.3 p_{sky}$ ) of the unit cell, respectively. The bottom inset in (c) shows the magnetic field distribution of the pure TD at 575 nm, which is strongly confined in the disk region (enclosed by the green-dashed line) with a vortex pattern (white arrows for field vectors).

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Moreover, the radially polarized feature of an electric-type OSL also makes it suitable for exploring some plasmonic dark modes in metal nanoparticles. Here, we firstly demonstrate the efficient and exclusive excitation of a plasmonic breathing mode in a planar metal nanoparticle, for example, a thin silver nanodisk placed in an electric-type OSL. Ideal plasmonic breathing modes in flat plasmonic nanoparticles are considered to be optically dark since they have zero net electric dipole moments. Plasmonic breathing modes therefore are more frequently studied by near-field spectroscopies, typically the electron energy loss spectroscopy (EELS) [51,52]. Figure 6(a) shows the calculated electron energy loss probability $Γ_{eels}$ of a silver disk with diameter 300 nm and thickness 30 nm. In the calculation, the silver disk is free standing in air and excited by an electron beam penetrating the disk through the center (see Appendix C for the calculation method). A single EELS peak is observed at ~420 nm, which can be identified as the fundamental breathing mode by its unique surface charge distribution featured with one circular and no radial node line of neutral charge. We also notice that it is possible to probe the plasmonic breathing modes by oblique incident plane wave with the aid of retardation effect [53]. However, the breathing mode (film mode) excited by a plane wave is mixed with other multipole edge modes, resulting in imperfectly symmetric surface charge distribution.

Fig. 6 (a) EELS spectrum of a silver nanodisk excited by an electron beam penetrating the disk through the center. The lower inset shows the excitation configuration. The upper inset shows the snapshot of the surface charge distribution of the fundamental plasmonic breathing mode. (b) Total and multipole decomposed SCS spectra of the silver nanodisk placed at the center ( $s = 0$ , see upper-right inset) of the unit cell of the electric-type OSL. (c) EELS spectrum of a gold nanorod homodimer excited by an electron beam passing through the center of the dimer gap as shown by the lower inset. (d) SCS spectra of the gold nanorod dimer placed at the center ( $s = 0$ , see lower-right inset) of the unit cell of the electric-type OSL. Surface charge distributions of the plasmon modes are shown by the insets close to the respect scattering peaks.

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Then, we put the silver nanodisk at the center of the unit cell of the electric-type OSL. The corresponding total and multipole decomposed SCS spectra are shown in Fig. 6(b). Interestingly, only one scattering peak is observed in the total SCS in the whole wavelength range. This mode is unambiguously confirmed as the fundamental plasmonic breathing mode as its surface charge distribution exhibits a perfectly radial symmetry (see the inset at the resonance peak). Compared to the EELS spectrum in Fig. 6(a), it is seen that the resonant energy of the breathing mode in Fig. 6(b) red-shifts to 520 nm because of the coupling effect between the particle and its image in the substrate. Such exclusive excitation of the plasmonic breathing mode benefits from the radially polarized in-plane electric field of an electric-type OSL as discussed in Fig. 2(b). When the silver nanodisk is placed at the unit cell center, the free electrons are resonantly driven by the radially polarized in-plane electric field, leading to the collective oscillation along the radial direction. But the excitation of other edge modes such as ED and EQ are prohibited due to the strong symmetry mismatch between their polarization configurations and the excitation field. As a matter of fact, the plasmonic breathing mode can be efficiently excited as long as the silver disk is placed closed to the center of the unit cell, although other edge modes can also be excited with relatively weaker strengths. The strong near field of plasmonic breathing modes render them high optical mode density and significant plasmon hybridization effect, which is crucial for surface-enhanced spectroscopies and optical sensing [54].

In addition to plasmonic breathing modes in single flat metal nanoparticles, hybridized plasmonic dark modes with zero net ED moments exist in coupled metal nanoparticles. A pedagogical example of the hybridized dark modes is the longitudinal anti-bonding dipolar mode in a plasmonic homodimer composed of two equivalent nanoparticles. Based on the plasmon hybridization model, such longitudinal anti-bonding dipolar mode is attributed to the out-of phase coupling of the EDs in the constitute particles. The longitudinal anti-bonding dipolar mode can be excited by electron beams passing through the dimer center [55]. For simplicity, here we consider a homodimer composed of two gold nanorods with a 10 nm gap between them. Each nanorod is 120 nm in length and 40 nm in diameter. Figure 6(c) shows EELS spectrum of the dimer free standing in air when the electron beam passes through the gap center, as sketched in the lower inset. The resonant peak at ~600 nm in the EELS spectrum can be identified as the longitudinal anti-bonding dipolar mode by its mirror symmetric surface charge distribution with respect to the dimer center (see the upper inset). Putting this dimer at the center of one unit cell center of the electric-type OSL, two scattering peaks are seen in the total SCS spectrum in the wavelength range 400 nm to 900 nm as shown in Fig. 6(d). For the broad scattering peak at shorter wavelength (520 nm), the surface charge distribution is of mirror symmetry with respect to the dimer center and has horizontal node lines of neutral charges. This mode is actually a transverse anti-bonding dipolar mode excited by the out-of plane electric field ( $E_{z}$ ) of the OSL. Notice that the transverse anti-bonding dipolar mode belongs to bright mode due to the constructive dipole moment. Instead, the longitudinal anti-bonding dipolar mode red-shifts to longer wavelength (660 nm) due to the presence of the substrate and shows a narrow scattering peak with high quality factor.

Finally, we stress that the above discussions are mainly concentrated on the electric-type OSL. Apparently, the electric and magnetic field configurations shown in Fig. 2 exchange for a magnetic-type OSL. As a result, a magnetic-type OSL has polarization features similar to an azimuthally polarized structured field. The interaction between a nanoparticle and a magnetic-type OSL must be different to that with an electric-type OSL. For example, when the Si sphere is placed at the center of a magnetic-type OSL, only the magnetic multipoles can be excited. Regarding the selective excitation of optical resonances, electric-type (magnetic-type) OSLs show similar functions to tightly focused radially (azimuthally) polarized vector beams [27–29]. However, we should note that the in-plane and out-of-plane field components of an OSL are in phase because of its evanescent nature. This is quite different to propagating structured fields, for example a tightly focused radially (azimuthally) polarized beam of which the corresponding phase difference is $π / 2$ at the focus. Such phase difference is crucial for tuning the scattering direction of nanoparticles [40], but the relevant topic is beyond the scope of this work. In addition, an ideal OSL has peculiar lattice symmetry which guarantees an integer topological invariant. It has been demonstrated that OSLs are robust to small losses provided that the real part of the field satisfies the lattice symmetry given by Eq. (2) and its magnitude is much larger than the imaginary part [34]. However, the required lattice symmetry to form an OSL will be broken if the phases between the evanescent waves are misaligned or the angle between the propagating directions of the adjacent pairs of standing waves deviates from $π / 3$ . The multipole responses of a nanoparticle excited by the interference field, of course, are also sensitive to these factors depending on how serious the polarization symmetry is broken. For example, if the phase misalignment between the TM polarized standing waves is not so large, the radial component of the in-plane field of the interference field will still be predominant, thus electric multipoles are more efficiently excited when a nanoparticle is placed at the center of the lattice.

3. Conclusions

In summary, we have shown that OSLs represent a kind of structured near-field which is able to explore rich optical resonances in nanoparticles. Depending on the position of nanoparticles placed in the OSL, electric and magnetic multipole modes can be selectively excited. In particular, the near-field configuration of an electric-type OSL has similar features with that of a radially polarized light, enabling efficient excitation of toroidal and plasmonic dark modes. Experimental generation of OSLs has been realized by interfering surface plasmon polaritons at metal surface. Alternatively, OSLs can also be generated at a dielectric surface through total internal reflection with delicate optical setups. The scattering responses of nanoparticles then can be investigated by far-field spectroscopies. Our results may promote applications of OSLs in optics such as on-chip optical signal manipulation, light emission control, near-to-far field conversion and nanoparticle trapping, etc.

Appendix A SCS calculation under plane waves and OSLs excitations

All the numerical results are calculated with the commercial electromagnetic simulation package COMSOL Multiphysics V5.2. The permittivity of gold and silver is taken from the empirical data [56]. A spherical calculation domain with perfectly matched layer being the boundary condition is adopted in the simulations. For the plane wave excitation, the whole calculation domain is simply set as air (refractive index 1) and a uniform background field is defined. For the OSL excitation, the calculation domain is divided into the upper air and lower glass regions with refractive index 1 and 1.5, respectively. Based on the discussion in Fig. 1, the background field to generate the OSL is the superposition of six incident and reflected waves in the glass region, and transmitted waves in the air region. The SCS of the nanoparticle excited by the background fields (plane wave or OSL) is calculated by $σ_{scs} = 2 Z_{0} \underset{A}{∯} P_{s c a t} \cdot \hat{n} d a / (\sqrt{ε_{b}} N^{2} E_{0}^{2})$ , where $A$ is a surface enclosing the nanoparticle, $P_{scat}$ is the scattered power flow, $\hat{n}$ is the unit normal vector of the surface, $Z_{0}$ is the vacuum impedance, $E_{0}$ is a constant electric field amplitude of the incident plane waves, $\sqrt{ε_{b}}$ is the permittivity of the background media (air for plane wave excitation and glass for OSL excitation), and $N$ is the number of incident plane waves ( $N = 1$ for plane wave excitation and $N = 6$ for OSL excitation).

Appendix B Cartesian multipole decomposition of SCSs in free space

The multipole moments in Cartesian coordinate system are determined by the polarization current $J (r^{'}) = - j ω ε_{0} (ε_{r} - 1) E (r^{'})$ induced in the nanoparticle, where $r^{'}$ is the position vector respect to the mass center of the particle and $ε_{r}$ is the relative permittivity of the nanoparticle, and $E (r^{'})$ is the electric field inside the nanoparticle. Then the electric dipole moment $P$ , magnetic dipole moment $M$ , toroidal dipole moment $T$ , electric quadrupole tensor ${\hat{Q}}^{e}$ and magnetic quadrupole tensor ${\hat{Q}}_{}^{m}$ are given by Eqs. (4)–(8) [44–46]:

P = - \frac{1}{j ω} \int J (r^{'}) d^{3} r^{'},

M = \frac{1}{2 c} \int r^{'} \times J (r^{'}) d^{3} r^{'},

T = \frac{1}{10 c} \int [[r^{'} \cdot J (r^{'})] r^{'} - 2 {r^{'}}^{2} J (r^{'})] d^{3} r^{'},

{\hat{Q}}^{e} = - \frac{1}{i ω} \int [r^{'} J (r^{'}) + J (r^{'}) r^{'} - \frac{2}{3} \hat{I} [r \cdot J (r^{'})]] d^{3} r^{'},

{\hat{Q}}^{m} = \frac{1}{3 c} \int [r^{'} \times J (r^{'})] r^{'} + r^{'} [r^{'} \times J (r^{'})] d^{3} r^{'},

where

\hat{I}

in Eq. (7) is a

3 \times 3

unit tensor. The SCS of the above multipoles then can be obtained as

ω^{4} Z_{0} {| P |}^{2} / (6 π ε_{0} c^{3} \sqrt{ε_{b}} N^{2} E_{0}^{2})

for ED,

ω^{4} Z_{0} {| M |}^{2} / (6 π ε_{0} c^{3} \sqrt{ε_{b}} N^{2} E_{0}^{2})

for MD,

ω^{4} Z_{0} {| P + i k T |}^{2} / (6 π ε_{0} c^{3} \sqrt{ε_{b}} N^{2} E_{0}^{2})

for TED,

ω^{6} Z_{0} {| {\hat{Q}}^{e} |}^{2} / (80 π ε_{0} c^{5} \sqrt{ε_{b}} N^{2} E_{0}^{2})

for EQ, and

ω^{6} Z_{0} {| {\hat{Q}}_{}^{m} |}^{2} / (80 π ε_{0} c^{5} \sqrt{ε_{b}} N^{2} E_{0}^{2})

for MQ.

Appendix C Calculation of electron energy loss probability

Numerically, the electron energy loss probability $Γ_{EELS} (ω)$ for an electron beam along $z$ direction can then be calculated by a line integration as [55]:

Γ_{EELS} (ω) = \frac{e}{π ℏ ω} \int d z Re [E_{z}^{ind} (ω, z) \exp (- i ω z / v)],

where

e

is the charge of the electron,

v

the relativistic speed of the electron,

ℏ

the reduced Planck constant,

ω

the angular frequency,

E_{z}^{ind} (ω, z)

is the z-component of the induced electric field

E^{ind}

which is calculated by a two-step study. In the first step, the radiation field of the electron beam in vacuum (without any particle) is obtained by calculating the full field of a z-direction line current source with the current density

j (z, ω) = - e \hat{z} \cdot \exp (i ω z / v)

. Then the obtained radiation field of the electron beam in the first step is used as a background field

E_{b}

in the second step to calculate the scattered field by the particle, i.e. the induced field

E^{ind}

.

Funding

Shenzhen Municipal Science and Technology Plan (JCYJ20170811154119292); National Natural Science Foundation of Guangdong (2015A030313748); China Postdoctoral Science Foundation (2018M630356, 2017M621274); National Natural Science Foundation of China (11672090).

Acknowledgments

Qiang Zhang thanks S. Tsesses and G. Bartal for their help in understanding the property of OSLs.

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Exploring optical resonances of nanoparticles excited by optical Skyrmion lattices

Abstract

1. Introduction

2. Results and discussion

2.1 Generation of OSLs based on total internal reflection

2.2 Field configuration of electric-type OSL

2.3 Selective excitation of electromagnetic multipoles by OSLs

2.4 Efficient excitation of TD and plasmonic dark modes by electric-type OSL

3. Conclusions

Appendix A SCS calculation under plane waves and OSLs excitations

Appendix B Cartesian multipole decomposition of SCSs in free space

Appendix C Calculation of electron energy loss probability

Funding

Acknowledgments

References

Cited By

Figures (6)

Equations (9)

Optics Express