Enhanced broadband spin Hall effects by core-shell nanoparticles

R. Shi; D. L. Gao; H. Hu; Y. Q. Wang; L. Gao

doi:10.1364/OE.27.004808

1. Introduction

Light can possess angular momentum, including spin angular momentum (SAM) and orbital angular momentum (OAM). Strong coupling of SAM and OAM, i.e., spin-orbit interaction (SOI), is a universal phenomenon in optics, which is underpinned by the spin properties of Maxwell’s equations [1]. SOI of light is associated with many polarization effects [2–5], with the spin Hall effect most attractive one [6–8]. Spin Hall effect of light (SHEL) is an analogy of electron applied with an electric field, which has the refractive index gradient acting as the electric field and the light polarization as the electron spin. For nanoparticles, the scattering of light [9,10] can give rise to SHEL. Although SHEL is usually weak in geometrical optics, with the development of nanotechnology, it becomes increasingly important in modern optics. The SOI-induced aberrations are not negligible and must be determined for precision optical devices. For instance, wavelength-scale image shifts are found in an imaging system due to the SOI of light [11]. Such systematic errors are comparable to the size of scatter and can bring undesirable damage to the resolution of imaging system.

On the other hand, SOI-induced spin Hall (SH) shift can be exploited for precision metrology in nanoscale, such as identifying graphene layer numbers and thickness by weak measurements [12–14], reconstructing vectorial field with subwavelength resolution [15], and mapping spin-dependent potential landscapes [16]. Very recently, spin Hall effects and transverse directional scattering by tightly focused light beam are used to sense angstrom displacement of nanoparticle [17,18]. The SH shifts are associated with two types of geometric phases, i.e., the Rytov-Vladimirskii-Berry phase and the Pancharatnam-Berry phase [19]. These geometric phases could be used to explain optical spin Hall effects in birefringent metamaterial [20], plasmonic chains [21] and other nanoscale structures [22]. However, the SH shift is usually trivial for a small nanoparticle (the maximal shift is about 0.3λ for a dipole) [23], and the detection of SH shift may require quantum weak measurements [24,25]. Hence, considerable enhancement and precise knowledge of the SH shift are of utmost significance for high-resolution imaging techniques. Recently, we demonstrate that SH shift can be effectively enhanced for nanoparticles with dual symmetry [26]. The enhanced SH shift is increased by one order of the maximal shift of a dipole, which is due to the coupling of electric and magnetic dipolar modes. However, for a single nonmagnetic nanosphere, the exciting of magnetic response relies on its dielectric permittivity and size parameter q (q = 2πR/λ) [27]. Once the size of the nanosphere is given, the incident wavelength that satisfies dual symmetry condition [28] is fixed.

In this paper, we propose a core-shell nanostructure to obtain broadband and tunable SH shift. The core-shell nanostructures are made of Mie spherical particles with either metallic core and dielectric shell or dielectric core and metallic shell. By using an appropriate radius ratio of the core-shell nanoparticle, broadband overlapping of the electric and magnetic dipolar modes (a₁ = b₁) can be realized in the infrared spectrum, where the SH shift is enhanced as well. What’s more, we find higher electric mode (a₂) can also be used to couple with electric dipolar mode (a₁) giving rise to double enhanced SH shift. Both far-field scattering and near-field patterns of the core-shell nanoparticle show that the enhanced SH shift is due to the strong SOI of light. Our proposed structure may pave the way towards the far-field superresolution microscopy [29] and nano-mechanical measurement [30–32].

2. Theoretical formulations

In this section, a scattering of a core-shell nanoparticle by circularly polarized light is considered, as is illustrated in Fig. 1. Without loss of generality, we consider a nanoparticle with radius of core a and shell b is illuminated by a left-hand circularly polarized (LCP) light. The time dependence $\exp (- i ω t)$ is suppressed. The LCP has the form $E_{i}^{L} = E_{0} e^{i k z} (\hat{x} - i \hat{y})$ . This circularly polarized wave can be treated as a superposition of x-polarized wave and y-polarized wave with a quadrature phase shift $Δ ϕ = 2 / π$ . Based on Mie theory [33], the incident LCP can be expanded in vector spherical harmonics,

E_{i}^{L} = \sum_{n = 1}^{\infty} E_{n} (M_{o m n}^{(1)} - i N_{e m n}^{(1)} - i M_{e m n}^{(1)} + N_{o m n}^{(1)})

H_{i}^{L} = \frac{- k}{ω μ} \sum_{n = 1}^{\infty} E_{n} (M_{e m n}^{(1)} + i N_{o m n}^{(1)} + i M_{o m n}^{(1)} + N_{e m n}^{(1)}),

where

E_{n} = E_{0} i^{n} (2 n + 1) / (n (n + 1))

,

M_{o m n}^{(1)}

,

N_{e m n}^{(1)}

,

M_{e m n}^{(1)}

,

N_{o m n}^{(1)}

are the vector spherical harmonics. The separation constant m = 1 and the superscript (1) to vector spherical harmonics means the radial dependence of the generating functions is the spherical Bessel function of the first kind [33].

Fig. 1 Illustration of spin Hall shift of scattered light from a core-shell nanoparticle. The spin Hall shift $Δ_{S H}$ (the red arrowed line) is perpendicular to the scattering plane. Due to the spin Hall effects, a far-field detector will assume the scattered light comes from the transversely displaced image, not from the real position.

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The scattered field can be obtained by imposing boundary conditions at the interface of core and shell (r = a) and at the interface of the shell and surrounding medium (r = b),

(E_{s h e l l}^{L} - E_{c o r e}^{L}) \times {\hat{e}}_{r} = (H_{s h e l l}^{L} - H_{c o r e}^{L}) \times {\hat{e}}_{r} = 0

(E_{i}^{L} + E_{s}^{L} - E_{s h e l l}^{L}) \times {\hat{e}}_{r} = (H_{i}^{L} + H_{s}^{L} - H_{s h e l l}^{L}) \times {\hat{e}}_{r} = 0.

The scattering fields are

E_{s}^{L} = \sum_{n = 1}^{\infty} E_{n} (i a_{n} N_{e m n}^{(3)} - b_{n} M_{o m n}^{(3)} - a_{n} N_{o m n}^{(3)} + i b_{n} M_{e m n}^{(3)})

H_{s}^{L} = \frac{k}{ω μ} \sum_{n = 1}^{\infty} E_{n} (i b_{n} N_{o m n}^{(3)} + a_{n} M_{e m n}^{(3)} + b_{n} N_{e m n}^{(3)} + i a_{n} M_{o m n}^{(3)}),

where a_n, b_n are the scattering coefficients of the core-shell particle, the superscript (3) means the radial dependence of the generating functions is the spherical Hankel function. The SH shift is defined as the transverse shift in perceived far-field location [34].

Δ_{S H} = \lim_{r \to \infty} r (S_{ϕ} / | S_{r} |) \hat{ϕ}

Where

S_{ϕ}

and

S_{r}

are the azimuthal and radial components of the scattered Poynting vector

S = (1 / 2) * E_{s}^{L} \times H_{s}^{L}

.

In what follows, we will qualitatively examine the SH shift for some typical cases, such as dipole, overlapping of electric and magnetic dipoles, and overlapping of dipoles with higher modes. Based on the above Mie theory, the Eq. (7) for SH shift can be written as [34]

Δ_{S H} = \frac{σ}{k} (\frac{| Re (S_{1}^{*} [\sum_{n = 1}^{\infty} (2 n + 1) a_{n} π_{n}] + S_{2} {[\sum_{n = 1}^{\infty} (2 n + 1) b_{n} π_{n}]}^{*}) |}{| S_{1} |^{2} + | S_{2} |^{2}}) \sin θ

Where

σ = \pm 1

for an incident wave of LCP and RCP states, S₁ and S₂ are the elements of the amplitude scattering matrix [33],

S_{1} (\cos θ) = \sum_{n = 1}^{\infty} \frac{2 n + 1}{n (n + 1)} [a_{n} π_{n} (\cos θ) + b_{n} τ_{n} (\cos θ)]

S_{2} (\cos θ) = \sum_{n = 1}^{\infty} \frac{2 n + 1}{n (n + 1)} [a_{n} τ_{n} (\cos θ) + b_{n} π_{n} (\cos θ)] .

In above equations, the angle-dependent functions $π_{n} (\cos θ)$ and $τ_{n} (\cos θ)$ are defined as, $π_{n} (\cos θ) = P_{n}^{1} (\cos θ) / \sin θ$ , $τ_{n} (c o s θ) = d P_{n}^{1} (\cos θ) / d θ$ . They can be calculated by their upward recurrent relations [33], and the first three expressions are $π_{0} = 0$ , $π_{1} = 1$ , $π_{2} = 3 \cos θ$ , and $τ_{1} = \cos θ$ , $τ_{2} = 3 \cos 2 θ$ .

3. Analytical discussions

3.1 Small particle with electric dipole

For a small particle with only electric dipole term a₁, the SH shift is reduced to $Δ_{S H} = (σ / k) [2 s i n θ / (1 + \cos^{2} θ)]$ [23], the shift solely depends on the scattering polar angles and reaches its maximum at $θ = π / 2$ where the transformation of SAM to OAM is complete [6].

3.2 Overlap of electric and magnetic dipoles

For a magneto-dielectric particle, the magnetic dipole b₁ will arise and interact with electric dipole a₁. Particularly, resonant SH shifts are demonstrated in systems with dual (a₁≈b₁) or anti-dual symmetry (a₁≈-b₁) [26]. The resonant behaviors can also be understood by investigating the denominator of Eq. (8), $D = | S_{1} |^{2} + | S_{2} |^{2}$ .

We assume that the magneto-dielectric particle excites only electric and magnetic dipoles, neglecting other higher modes. Then the denominator D has form,

D \propto {| a_{1} + b_{1} \cos θ |}^{2} + {| a_{1} \cos θ + b_{1} |}^{2} .

It is evident that when the particle has the dual symmetry (a₁≈b₁), D is close to zero around

θ \approx π

. Thus, the SH shift will be enhanced at the backward scattering. The situation is the same for an anti-dual system as well, i.e., the resonant shift appears at the forward scattering.

3.3 Overlap of electric dipole and electric quadrupole

With the increase of particle’s size, higher modes (quadrupole, octupole, et al.) come to play, resulting in the interacting of dipole and higher-order modes. For a plasmonic particle with low dissipative loss, the magnitude of electric quadrupole is of the same order of that electric dipole [35], whereas the magnetic modes are negligible.

The denominator D is as follows,

D \propto | 3 a_{1} + 5 a_{2} \cos θ |^{2} + | 3 a_{1} \cos θ + 5 a_{2} \cos 2 θ |^{2} .

When the electric dipole a₁ and the quadrupole a₂ have the same magnitudes and oscillate in phase (a₁ = a₂), the first term in the right side of Eq. (12) goes to zero at $θ = 127 °$ ; While the second term goes to zero at $θ = 55 °$ and $θ = 151 °$ . Hence, one may expect resonant SH shift around $θ = 140 °$ . When a₁ and a₂ oscillate out of phase (a₁ = -a₂), the first term in the right side of Eq. (12) goes to zero at $θ = 53 °$ ; and the second term goes to zero at $θ = 30 °$ and $θ = 125 °$ . SH shift will be enhanced around $θ = 40 °$ .

As we will demonstrate in the following section, those characteristics of overlapping multiple modes play an essential role in broadening the range of resonant SH shift.

4. Numerical demonstrations

4.1 Broadband enhanced SH shift by overlapping electric and magnetic dipoles

SH shift can be effectively enhanced in a dual system [26], where electric dipole and magnetic dipole are equally excited. However, the excitement of magnetic dipole relies on the magneto-dielectric media or high refractive index. While magneto-dielectric particles are rare in nature. And for a single spherical particle with high refractive index, the simultaneous exciting of electric and magnetic dipoles only happens in a very narrow spectral range. Then, one question arises: can we manifest enhanced SH shift in a broad spectral range, while maintaining the particle’s size? Here, we shall show that a core-shell spherical particle can overlap its electric and magnetic dipoles in rather broadband spectrum.

In Fig. 2(a), we demonstrate the scattering coefficients of electric and magnetic for both dipoles (a₁ and b₁) and quadrupoles (a₂ and b₂). The quadrupole modes and higher order modes are small and negligible in the wavelength range larger than 1200 nm. The electric dipole and magnetic dipole overlap well with each other in near-infrared spectra from λ = 1.22 μm to λ = 1.31 μm, where the total scattering efficiency reaches resonances [36]. In this resonant range, the electric dipole comes from the localized surface plasmon in the metallic core [37], and the magnetic dipole comes from the optically-induced cavity mode of the dielectric shell [38,39]. Interestingly, the overlapping range is also corresponding to the dual behavior of the core-shell system. In a system with dual symmetry, the scattering light preserves its helicity after scattered in the system [28]. This scattering behavior could be characterized by using the transfer function T, which is defined as the ratio between the energy scattered with opposite polarization and the energy scattered in the same polarization with the incident light [40]. As is shown in Fig. 2(a), the transfer function T tends to zero in the overlapping range, meaning the core-shell particle acts as a dual particle, and almost all the scattered lights are in the same polarization with the incident light. As demonstrated in our previous work [26], spin-orbit interaction, as well as the associated spin Hall effect of light, can be enhanced in a dual system due to the interference of electric and magnetic dipoles. While for core-shell particles, the enhanced SH shifts have a rather broadband range from λ = 1.22 μm to λ = 1.31 μm, as is shown in Fig. 2(b). According to Eq. (11), the maximal value of shift is located around $θ \approx π$ for a given incident wavelength. The SH shift can be enhanced up to two times of the incident wavelength, large enough to be detected in experiment.

Fig. 2 (a) Norm of the Mie scattering coefficients (left horizontal axis) and transfer function T (right horizontal axis) versus the incident wavelength. (b) Contour plot of spin Hall shifts $Δ_{S H}$ as a function of the incident wavelength and scattering angle. $Δ_{S H}$ are enhanced up to two times of the incident wavelength, and show robust to the wavelength and scattering angle where the electric and magnetic dipoles are overlapped. The core-shell nanoparticle consists of a silver core (with radius a = 68 nm) and a dielectric shell (with radius b = 250 nm and the refractive index of 2.5).

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The enhancement of spin-orbit interaction in the dual core-shell nanoparticle is also illustrated via circular polarization of the near field. For a pure state of incident circularly polarized light, the circular polarization degree (CPD) is 1, which CPD is defined as $C = | ε_{0} E^{*} \times E + μ_{0} H^{*} \times H | / (ε_{0} | E |^{2} + μ_{0} | H |^{2})$ [41]. And the minimal CPD is 0, i.e., the field is linear polarized, meaning the spin angular momentum is completely transformed to orbital angular momentum. As is shown in Fig. 3(a) and 3(b), when the electric and magnetic dipole are simultaneously excited in the dual range (from λ = 1.22 μm to λ = 1.31 μm), the electric and magnetic field intensities are greatly enhanced at the shell of the particles. The enhancement of magnetic dipole response is due to the coupling of incident light with the circular displacement currents inside the particle’s shell [27]. For a nonmagnetic structure with high refractive index, the generation of magnetic responses requires that the wavelength inside the particle’s shell (λ/n≈1250 nm/2.5 = 500 nm) is comparable to the particle’s diameter (2b = 500 nm). The interaction of equal strengths of electric and magnetic field results in the strong spin-orbit interaction near the core-shell nanoparticle, as demonstrated the two linear polarized regions in Fig. 3(c). This near-field spin-orbit interaction is associated with the enhanced spin Hall shift in the far field.

Fig. 3 Near-field distributions of a core-shell nanoparticle with dual behavior when the electric and magnetic dipoles have equal strength and oscillate in phase. (a) Normalized electric field. (b) Normalized magnetic field. (c) Circular polarization degree (CPD). The blue regions in (c) indicate the field is linear polarization as the result of strong spin-orbit interaction. The incident wavelength is 1250 nm, and other parameters are the same as those in Fig. 2.

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4.2 Double enhanced SH shift by overlapping electric dipole and electric quadrupole

When the incident wavelength goes to the visible range, higher modes will be induced and overlapped up the electric dipole mode. According to Eq. (12), the SH shifts will be resonant when the electric dipole a₁ and quadrupole a₂ have equal magnitude and oscillate in phase or out of phase. It indicates that the SH shifts can be tuned by manipulating the magnitudes and phase between a₁ and a₂. As is shown in Fig. 4(a), quadrupole modes arise in the range of 650 nm to 700 nm. The inset shows the strength of interference of dipole and quadrupole modes and the phase difference between these two modes. The strength of destructive interference begins to increase at around 675 nm, where the phase difference gradually changes to π. To one’s interest, the reversal of phase differences between plasmon modes is usually used to explain the reversal of signs of Goos-Hänchen shift [34]. In our case, the interaction could give rise to an additional enhancement of SH shift. Figure 4(b) shows the emergence of extra enhancement at a small scattering angle (around $θ = 40 °$ ) due to the destructive interference of a₁ and a₂. With the decrease of interference intensity beyond λ = 680 nm, the SH shifts decrease as well.

Fig. 4 (a) Norm of the Mie scattering coefficients and (b) spin Hall shifts for a core-shell nanoparticle with a dielectric core (radius a = 98 nm and refractive index of 1.5) and silver shell (with radius b = 105 nm). Extra enhanced spin Hall shifts emerge at the forward direction, where the electric quadrupole modes (a₂) couple with electric dipole modes (a₁). The inset shows the interference strength and phase differences of a₁ and a₂.

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It is well-known that SH shift is due to the spin-orbit interaction of light [6]. In experiment, diattenuation d(θ) is an excellent indicator to examine the spin-orbit interaction from far field, which is defined as the differential attenuation of orthogonal polarization states [34]. The value of d(θ) is corresponding to the transformation of SAM to OAM, i.e., d(θ) = 0 for no transformation and d(θ) = 1 for complete transformation. Figure 5 shows the SH shifts and corresponding diattenuations with the variation of scattering angle. It is clearly shown that SH shifts are enhanced at both small scattering angles and large scattering angles in the range of 600 nm to 700 nm. And the corresponding diattenuations (Fig. 5(b)) reveal that the enhancement of SH shift origins from the complete transformation of SAM to OAM (the grey region with |d(θ)| = 1). Meanwhile, the near-field distributions of CPD and Poynting vectors help us to understand the double enhanced SH shift. As is shown in Fig. 6(a), both the CPD and Poynting vector shows a typical electric quadrupole distribution, indicating that the quadrupole mode involves in the transformation of SAM to OAM and contributes the extra resonant of spin Hall shift at the small scattering angles. While at a longer wavelength (λ = 750 nm, see Fig. 6(b)), the field distributions show dipole like patterns, which indicates that there is only dipole interaction for this case. This can also be verified by the scattering coefficients in Fig. 4(a), where the dipole mode is dominant, and the quadrupole mode is negligible.

Fig. 5 Contour plot of (a) the spin Hall shifts and (b) the corresponding diattenuation. The regions of enhanced spin Hall shifts coincide with that (gray region in (b)) of its diattenuation, where the full transformation of spin angular momentum to orbital angular momentum takes place. The core-shell nanoparticle’s parameters are the same as those in Fig. 4.

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Fig. 6 Field distributions for the cases of (a, c) quadrupole resonance and (b, d) dipole resonance. The patterns of circular polarization degree have similar characteristics with the corresponding Poynting vectors. The core-shell nanoparticle’s parameters are the same as those in Fig. 4.

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Note that, Fano resonance also arises when the board electric dipole modes interfere with the narrow electric quadrupole modes in the core-shell nanoparticle. The scattering coefficients of a₁ and a₂ in Fig. 4(a) clearly show the origin of the Fano resonance. Figure 7 is the backward and forward scattering of the particle. Around the Fano resonance (about λ = 675 nm), the scattering cross-section exhibits asymmetric shape. Meanwhile, a typical flip of backward and forward scattering happens as well. As is shown in Fig. 7(b), the overall scattering changes from mainly forward scattering to mainly backward scattering. Around the Fano resonance (incident wavelengths are small than 675 nm), the forward scattering dominates and the resonance of SH shift happens at small scattering angles, which facilitates the observation of SH shift in experiments [42–44]. The inherent sensitivity of Fano resonances also have high potential for probing spin-orbit interaction [45]. What’s more, the scattering intensity is large enough to be detected in experiments at the angles of SH shift enhanced.

Fig. 7 (a) Forward (θ = 0°) and backward (θ = 180°) scattering efficiencies and (b) the normalized scattering intensities around the Fano resonance. The core-shell nanoparticle’s parameters are the same as those in Fig. 4.

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5. Conclusion

To conclude, we elucidated two mechanisms to obtain enhanced broadband spin Hall shifts, i.e., by overlapping electric and magnetic dipoles or by overlapping electric dipole and electric quadrupole. The coupling of different electric/magnetic modes can be tuned by the parameters of the core-shell nanoparticles. Both far-field scattering patterns and near-field distributions help us to understand the spin-orbit interaction of light. The enhanced spin Hall shifts by core-shell nanostructures are robust with respect to small variations of incident wavelength and scattering angles. Hence, the proposed nanoparticles can be used to detect local polarization of structured optical field via its spin Hall shifts. Collocation of electric and magnetic optical responses is possible with all-dielectric dimer [46], hence we anticipate that similar enhancement of spin Hall effect can be observed for multiple nanoparticles. In addition, a solid understanding of nanoparticle’s spin-orbit interaction with light will render us able to further improve the sensitivity of miniaturized mechanical sensors [31], which relies on accurate detection of the particle’s position.

Funding

National Natural Science Foundation of China (Grant Nos. 11504252 and 11774252), the National Science of Jiangsu Province (Grant Nos. BK20150306 and BK20161210), China Postdoctoral Science Foundation Grant (2018M630596), the Qing Lan project, “333” project (Grant No. BRA2015353), Collaborative Innovation Center of Suzhou Nano Science and Technology, and PAPD of Jiangsu Higher Education Institutions.

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Enhanced broadband spin Hall effects by core-shell nanoparticles

Abstract

1. Introduction

2. Theoretical formulations

3. Analytical discussions

3.1 Small particle with electric dipole

3.2 Overlap of electric and magnetic dipoles

3.3 Overlap of electric dipole and electric quadrupole

4. Numerical demonstrations

4.1 Broadband enhanced SH shift by overlapping electric and magnetic dipoles

4.2 Double enhanced SH shift by overlapping electric dipole and electric quadrupole

5. Conclusion

Funding

References

Cited By

Figures (7)

Equations (12)

Optics Express