Polarization characteristics and transverse spin of Mie scattering

Zhaolou Cao; Zhaolou Cao; Chunjie Zhai

doi:10.1364/OE.511898

1. Introduction

Polarization is one of the most important properties of an electromagnetic wave. While classical polarization optics usually studies a paraxial beam with a one-dimensional (1D) or two-dimensional (2D) polarization state, complicated polarization states of a structured beam have been a hot topic in recent years with the progress in nano-photonics. As it offers many more degrees of freedom to manipulate the topological distribution of polarization state than that of intensity and phase, various potential applications have been proposed exploiting the vectorial nature of an electromagnetic wave, e.g. it is possible to develop novel photonic devices [1–4], manipulate nanoparticles [5–7] and locate particles with subatomic precision [8] through modulating the polarization. Assisted by wavefront and polarization shaping techniques, it was shown that the polarization state features a subwavelength structure, such as a polarization Möbius strip [9], around the focus of a tightly focused beam.

To characterize complicated 3D polarization states, an unambiguous mathematical description of its true 3D features is necessary. Akin to the 2D paraxial beam, it was shown that a 3 × 3 polarization matrix could be utilized to describe a genuine 3D polarization state [10–12], which can be further decomposed into an incoherent superposition of three parts: a fully polarized state, a state composed by two uncorrelated orthonormal states and a 3D unpolarized state. 3D polarization states can be subsequently classified into regular and nonregular states according to the results of characteristic decomposition. Several works have been conducted to investigate the properties of 3D polarization states of different structured light fields, such as tightly focused beam [13], evanescent wave [14] and Gaussian Schell-model beam [15].

Meanwhile, light scattering by nanoparticles is of fundamental significance in a wide range of physics, such as astronomy, imaging, climate studies, etc. Despite it is an old topic, novel scattering phenomena are still emerging nowadays fostered by the flourishing field of photonics, leading to novel design of photonic devices, like nano-antennas and nano-routers for optics [16–19]. While the scattered light has a 2D polarization state (electric field oscillates in a direction perpendicular to the propagation direction) in the far field, it exhibits a complicated pattern in the near field due to the non-trivial radial component of electric field [20,21]. In this case, transverse spin existing in the focus of a tightly focused beam [22,23] can also be observed in the near field of light scattering [24,25]. To fully exploit the near-field light field, it is desirable to have a deep understanding of its polarization state. Despite the work on structured beams during the focusing or reflection, how will the polarization quantities be in the near field of scattered light still needs to be further investigated. To address this issue, we revisit the conventional Mie scattering problem in this work, focusing on the characterization of polarization state in the near field of partially polarized scattered light.

This work is organized as follows. In Sec. 2, the scattering problem and its solution are described. In Sec. 3, polarization quantities, including degree of nonregularity, polarization dimension, and spin are introduced. In Sec. 4, we present the results and discuss the behavior of the polarization quantities. Main findings are summarized in Sec. 5.

2. Light scattering problem

We consider the problem of a quasimonochromatic partially-polarized beam scattered by a nanosphere, schematic of which is illustrated in Fig. 1. The nanosphere located in a plane wave propagating in the z-direction scatters light in different directions and forms a complicated intensity pattern. Based on the rigorous Mie scattering theory, the electric field of scattered light for a spherical particle in an x-polarized plane wave can be readily represented as the sum of an infinite series, given by

(1)$${\textbf{E}_s} = \sum\limits_{m = 1}^\infty {{E_m}[{i{a_m}\textbf{N}_{el m}^{(3 )} - {b_m}\textbf{M}_{ol m}^{(3 )}} ]}, $$

in the spherical coordinate system following the work of Bohren and Huffman [26], where E_m = i^mE₀(2m + 1)/m(m + 1), E₀ denotes the electric field amplitude of the incident plane wave. a_m and b_m are conventional Mie scattering coefficients, expressed as

(2)$${a_m} = \frac{{\mu {n^2}{j_m}({nx} ){{[{x{j_m}(x )} ]}^\prime } - {\mu _1}{j_m}(x ){{[{nx{j_m}({nx} )} ]}^\prime }}}{{\mu {n^2}{j_m}({nx} ){{[{xh_m^{(1)}(x )} ]}^\prime } - {\mu _1}h_m^{(1)}(x ){{[{nx{j_m}({nx} )} ]}^\prime }}}, $$

(3)$${b_m} = \frac{{{\mu _1}{j_m}({nx} ){{[{x{j_m}(x )} ]}^\prime } - \mu {j_m}(x ){{[{nx{j_m}({nx} )} ]}^\prime }}}{{{\mu _1}{j_m}({nx} ){{[{xh_m^{(1)}(x )} ]}^\prime } - \mu h_m^{(1)}(x ){{[{nx{j_m}({nx} )} ]}^\prime }}}, $$

where j_m(x) and h_m⁽¹⁾(x) are respectively the spherical Bessel and spherical Hankel functions, n is the refractive index of nanosphere, x = 2πr/λ, r is the particle radius, λ is the illumination wavelength, μ and μ₁ are respectively the permeability of the surrounding medium and particle, [xf(x)]’ denotes the derivative of xf(x) with respect to x. The Mie scattering coefficients can be further transformed into the sum of a broadband background and a term associated with sharp resonance so that Mie scattering can be regarded as a cascade of fano resonances [27]. $\textbf{N}_{el m}^{(3 )}$ and $\textbf{M}_{ol m}^{(3 )}$ stand for vector spherical harmonics. Detailed expressions can be found in Ref. [26]. The mth term of electric field in the spherical coordinate system of (r, θ, φ) are expressed as

(4)$${E_r} = i{E_m}{a_m}\cos \varphi m({m + 1} )\sin \theta {\pi _m}({\cos \theta } )\frac{{h_m^{(1)}({kr} )}}{{kr}}, $$

(5)$${E_\theta } = i{E_m}{a_m}\cos \varphi {\tau _m}({\cos \theta } )\frac{{{{[{krh_m^{(1)}({kr} )} ]}^\prime }}}{{kr}} - {E_m}{b_m}\cos \varphi {\pi _m}({\cos \theta } )h_m^{(1)}({kr} ), $$

(6)$${E_\varphi } ={-} i{E_m}{a_m}\sin \varphi {\pi _m}({\cos \theta } )\frac{{{{[{krh_m^{(1)}({kr} )} ]}^\prime }}}{{kr}} + {E_m}{b_m}\sin \varphi {\tau _m}({\cos \theta } )h_m^{(1)}({kr} ), $$

where k = 2π/λ. Note that the time variable is omitted for the sake of brevity.

Fig. 1. Schematic of the scattering problem. PPB: partially-polarized beam, NS: nanosphere, SL: scattered light.

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For a nanosphere with a radius much smaller than the wavelength, |a_m| and |b_m| (m > 1) approach to zero. In this case, we simply need to consider the lowest-order coefficients a₁ and b₁ which actually denote separately the (scaled) electric and magnetic polarizabilities of electric (ED) and magnetic dipole (MD) moments. When the relative permeability of the nanosphere is roughly 1, |a₁| is usually much larger than |b₁| and it is the ED that dominates the electric field distribution.

The scattered light for the incidence of y-polarized plane wave can be obtained through rotating that of x-polarized beam in the spherical coordinate system, viz.

(7)$${\textbf{E}_{s,y}}({r,\theta ,\varphi } )= {\textbf{E}_{s,x}}({r,\theta ,\varphi - \pi /2} ), $$

where the subscripts x and y respectively denote the x- and y-polarized beams. When the incident plane wave is partially polarized, the scattered light can be regarded as a superposition of that of x and y components, respectively expressed as E_x(ω)E_s,x and E_y(ω)E_s,y, where E_x(E_y) is the complex field amplitude of the x-polarized (y-polarized) component of the incident beam, ω denotes the angular frequency. To numerically perform the superposition, it is necessary to determine the second-order polarization properties through the correlation coefficient, denoted as η, between the x- and y-polarized constituents, given by

(8)$$\eta = \frac{{\left\langle {E_x^\ast (\omega ){E_y}(\omega )} \right\rangle }}{{\sqrt {{w_x}} \sqrt {{w_y}} }}, $$

(9)$${w_x} = \left\langle {{{|{{E_x}(\omega )} |}^2}}\rangle \;\textrm{and}\; {w_y} = \left\langle {{{|{{E_y}(\omega )} |}^2}} \right .\right\rangle$$

where the angle brackets denote the average over an ensemble of monochromatic realizations of the electric field. In the limit of η = 1, the incident beam is fully polarized and the superposition of scattered light for the x- and y-polarized beams should be coherent. Note that although it is set to be real in this work, η can be a complex number theoretically.

3. Polarization characteristics

As mentioned in Section 1, the polarization characteristics of a three-dimensional random stationary light field at a fixed position r are fully determined by the 3 × 3 Hermitian polarization matrix that has the structure of a covariance matrix, given by

(10)$$\textbf{R}(\textbf{r} )= \left\langle {{\textbf{E}^\ast }(\textbf{r} ){\textbf{E}^T }(\textbf{r} )} \right\rangle = \left( {\begin{array}{ccc} {\left\langle {{{|{{E_r}} |}^2}} \right\rangle }&{\left\langle {{E_r}^\ast {E_\theta }} \right\rangle }&{\left\langle {{E_r}^\ast {E_\varphi }} \right\rangle }\\ {\left\langle {{E_\theta }^\ast {E_r}} \right\rangle }&{\left\langle {{{|{{E_\theta }} |}^2}} \right\rangle }&{\left\langle {{E_\theta }^\ast {E_\varphi }} \right\rangle }\\ {\left\langle {{E_\varphi }^\ast {E_r}} \right\rangle }&{\left\langle {{E_\varphi }^\ast {E_\theta }} \right\rangle }&{\left\langle {{{|{{E_\varphi }} |}^2}} \right\rangle } \end{array}} \right), $$

where the elements are the second-order moments of zero-mean electric field components, the asterisk and superscript T respectively stand for the complex conjugate and transpose operation. E(r) = [E_r(r), E_θ(r), E_φ(r)]^T is the electric field in the spherical coordinate system given by the Mie solution in Eqs. (4)–(6). The diagonal elements of R can be interpreted as the intensities associated with the respective r, θ and φ components of the electric field. By this way, the total intensity averaged over the measurement time is given by

(11)$$I = \textrm{trace}(\textbf{R})\left\langle {{{|{{E_r}} |}^2}} \right\rangle + \left\langle {{{|{{E_\theta }} |}^2}} \right\rangle + \left\langle {{{|{{E_\varphi }} |}^2}} \right\rangle .$$

Substituting Eq. (8) into Eq. (10), the expression of R can be obtained as

(12)$${R_{ij}} = {w_x}E_{\zeta x}^\ast {E_{\xi x}} + {w_y}E_{\zeta y}^\ast {E_{\xi y}} + \sqrt {{w_x}{w_y}} \eta E_{\zeta x}^\ast {E_{\xi y}} + \sqrt {{w_x}{w_y}} {\eta ^\ast }E_{\zeta y}^\ast {E_{\xi x}}, $$

where R_ij is the element of the polarization matrix, i (i = 1, 2, 3) and j (j = 1, 2, 3) respectively denote the row and column indices, the subscripts x and y respectively denote the variables for the x- and y-polarized incident light, ζ is defined to be r, θ, and φ respectively for i = 1, 2, and 3, ξ is the same as ζ but for j. E_ζx and E_ζy (ζ = r, θ, φ) are amplitudes of scattered light for the incidence of x- and y-polarized plane waves of unit intensity, respectively.

In order to have better understandings of the polarization matrix, it is natural to decompose the polarization matrix into simple forms with specific physical meanings to interpret the main features of 3D polarization state, where the 3D polarization state is regarded as a mixture of pure states [14]. As the Hermitian polarization matrix has three non-negative eigenvalues of λ₁, λ₂, and λ₃ (ordered as λ₁ ≥ λ₂ ≥ λ₃), it can always be expressed via the characteristic decomposition

(13)$$\textbf{R} = I[{{P_1}{{\hat{\textbf{R}}}_p} + ({{P_2} - {P_1}} ){{\hat{\textbf{R}}}_m} + ({1 - {P_2}} ){{\hat{\textbf{R}}}_u}} ], $$

where I is the intensity given by Eq. (11), P₁ = (λ₁ - λ₂)/I and P₂ = 1 - 3λ₃/I are defined as the indices of polarization purity,

(14)$${\hat{\textbf{R}}_p} = \textbf{U}\textrm{diag} ({1,0,0} ){\textbf{U}^{\dagger}}, $$

(15)$${\hat{\textbf{R}}_m} = \frac{1}{2}\textbf{U}\textrm{diag} ({1,1,0} ){\textbf{U}^{\dagger} }, $$

(16)$${\hat{\textbf{R}}_u} = \frac{1}{3}\textbf{I}, $$

where U is the unitary matrix diagonalizing the polarization matrix, I is the 3 × 3 identity matrix, the dagger denotes the conjugate transpose. The matrices ${\hat{\textbf{R}}_p}$ and ${\hat{\textbf{R}}_u}$ respectively represent a fully polarized state and unpolarized 3D state. ${\hat{\textbf{R}}_m}$ composed by two uncorrelated orthonormal states helps classify the 3D polarization states into regular and nonregular states. For regular states, ${\hat{\textbf{R}}_m}$ represents a 2D unpolarized state that can be considered an equiprobable mixture of two mutually orthogonal states with polarization ellipses lie in the same plane. In this case, the polarization ellipse of ${\hat{\textbf{R}}_m}$ evolves randomly but always remains in a fixed plane. For nonregular states, ${\hat{\textbf{R}}_m}$ corresponds to an incoherent mixture of two equal-intensity states with polarization ellipses confined to different planes. Except the special case of P₁ = P₂, a light field will be regular only if ${\hat{\textbf{R}}_m}$ is real valued. In other words, a complex-valued ${\hat{\textbf{R}}_m}$ corresponds to a non-regular polarization state.

A series of physically invariant polarization quantities [28] have been proposed to quantitatively characterize the non-regular polarization state, among which the degree of nonregularity, denoted as P_N, and polarization dimension, denoted as P_D, are employed in this work. The former indicates a measure for assessing the nonregularity of R, expressed as

(17)$${P_N} = 4({{P_2} - {P_1}} ){\hat{m}_3}, $$

where 0 ≤${\hat{m}_3}$≤ 1/4 is the smallest eigenvalue of the real part of ${\hat{\textbf{R}}_m}$. A regular polarization state requires P_N = 0, i.e., P₁ = P₂ or ${\hat{m}_3}$= 0, while 0 < P_N ≤ 1 is associated with nonregular polarization state. The extreme case of P_N = 1 is found exclusively for maximally nonregular states with P₁ = 0, P₂ = 1, and ${\hat{m}_3}$= 1/4.

The polarization dimension characterizes the intensity-distribution spread. It can be expressed as [29]

(18)$${P_D} = 3 - \sqrt {2[{{{3({\hat{\lambda }_1^2 + \hat{\lambda }_2^2 + \hat{\lambda }_3^2} )} / {{I^\textrm{2}}}} - 1} ]}, $$

where ${\hat{\lambda }_1}$, ${\hat{\lambda }_2}$, and ${\hat{\lambda }_3}$ are the eigenvalues of the real part of complex-valued R. P_D is bounded between 1 and 3. The lower limit of 1 takes place only for 1D linearly polarized light, while the upper limit of 3 is encountered for a 3D light field with isotropic intensity in three directions.

In the far field of Mie scattering, the light should be regular and the polarization dimension should be not greater than 2, as the polarization ellipse is always perpendicular to the propagation direction (the radial component of electric field is negligible). Unlike the far field, the non-zero radial component leads to a 3D polarized light field (P_D > 2) in the near field. Since the scattered light field is generated from the illumination of a partially polarized plane wave, the smallest eigenvalue λ₃ of R is always zero [11]. In this case, the light field does not include the component of completely unpolarized 3D state as P₂ = 1.

Recently, light beams carrying spin angular momentum (SAM) and orbit angular momentum (OAM) have attracted much attention. It is well known that at subwavelength scales, many optical phenomena, such as propagation, diffraction, and focusing, are affected by the SAM of light, e.g. the spin-momentum locking effect is observed in the near field of Mie scattering [25]. For a partially polarized beam, the SAM can be derived in terms of the elements of the polarization matrix, as it is related to the polarization state. Analogy to fully polarized states, the electric-field spin for an arbitrary mixed state in the spherical coordinate system can be obtained by averaging over time, given by [30]

(19)$$\textbf{n} = {\mathop{\rm Im}\nolimits} \left\langle {{\textbf{E}^\ast } \times \textbf{E}} \right\rangle = 2{\mathop{\rm Im}\nolimits} {\left[ {\begin{array}{ccc} {{R_{23}}}&{ - {R_{13}}}&{{R_{12}}} \end{array}} \right]^T }, $$

where Im[·] denotes the imaginary part. According to the characteristic decomposition, the spin vector of a partially polarized state is the vector sum of spin vectors of its pure polarization components.

4. Results

In this section, a numerical study concerning the polarization properties and spin structure of Mie scattering is presented. Since the scattered light pattern is complicated for a particle with a radius much larger than the wavelength, we focus on the scattering by nanospheres much smaller than the wavelength, where only the ED (a₁) and MD (b₁) need to be taken account of. Without loss of generality, a particle with a radius (r) of 20 nm, refractive index (n) of 3.5, and relative permeability (μ₁) of 1 is employed in the simulation. The incident plane wave propagates in the z direction with a wavelength (λ) of 500 nm. The intensity w_x and w_y directions are both set to be 1 except for explicit explanation.

As the radial component of light field is non-trivial in the near field, it has a significant impact on polarization characteristics. Intensity distribution of scattered light normalized by the maximum value for unpolarized and x-polarized plane waves is given in Fig. 2. The axisymmetric intensity is independent of φ for the unpolarized plane wave in Fig. 2(a), where the azimuthal component is always zero due to the axisymmetry. On the surface at r = 20 nm, the intensity has its maximum value at θ = π/2 and minimum value in the forward and backward direction due to the nonzero radial component. As the radial component attenuates quasi-exponentially with the propagation distance, the intensity distribution varies intensively only 1 µm away from the nanosphere, where the intensity is mainly contributed by the polar component and has the maximum value in the forward direction and minimum value at θ ≈ π/2. The slight asymmetry is caused by the appearance of nonzero MD; more precisely, it is a result of electric-magnetic dipole interference, despite |b₁| is much smaller than |a₁|. For the x-polarized plane wave, the intensity distribution is the same as that in Fig. 2(a) in the xOz plane. As both radial and polar components vanish, the intensity changes little during the propagation in the yOz plane. Since the x-polarized plane wave can be regarded as a coherent superposition of two plane waves with orthogonal polarization states, the different intensity patterns are direct manifestation of the influences of partial polarization.

Fig. 2. Normalized near-field intensity (a) for (w_x, w_y, η) = (1, 1, 0) and (b) in the yOz plane for (w_x, w_y, η) = (1, 0, 0).

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According to Eq. (17), P₁ and ${\hat{m}_3}$ are two intermediate variables that determine the degree of nonregularity. Figure 3 gives the two variables in the near field of scattered light. When an unpolarized plane wave is scattered, the radial component of the scattered light vanishes and the polar and azimuthal components have the same magnitude in the forward and backward direction. In this case, the scattered light behaves like a 2D unpolarized beam, which leads to zero P₁ and ${\hat{m}_3}$. In Fig. 3(a), it is observed that P₁ reaches near-zero minimum at r = 187.3 nm in the radial direction, indicating that the polarization matrix is dominated by the second term ${\hat{\textbf{R}}_m}$ in the region and the total polarization can be decomposed into two equal-intensity states with polarization ellipses in different planes. Analytical derivation (see Section S1 of Supplement 1) shows that the two non-zero eigenvalues of R are the same for an ED at k₀r = 2.354. Since the MD magnitude (|b₁|) is usually much smaller than the ED magnitude (|a₁|) for a dielectric nanosphere, the local minimum is always near r = 187.3 nm under the illumination with a wavelength of 500 nm regardless of the magnitude of a₁. In the polar direction, P₁ reaches the maximum value at θ = π/2 (see Section S2 of Supplement 1). By comparison, ${\hat{m}_3}$ is always zero in the forward and backward direction as the scattered light is 2D unpolarized, as shown in Fig. 3(b). If the MD is not taken account of, both R and real(R) will have a rank of 2 in the side direction as the polar component vanishes. Therefore, ${\hat{m}_3}$ reaches local minimum in the side direction. In Ref. [29], it is shown that ${\hat{m}_3}$ reaches the value of 1/4 when ${\hat{\textbf{R}}_m}$ is in a perfect nonregular state, interpreted as an equiprobable mixture of a circularly polarized state and a mutually orthogonal linearly polarized state. In this work, the condition can be interpreted as E_r = iE_θ and |E_r|² + |E_θ|² = |E_φ|², which cannot be strictly satisfied for the scattered light. Therefore, the maximum value of ${\hat{m}_3}$ for the scattered light cannot reach 1/4 in the entire space. Nevertheless, ${\hat{m}_3}$ reaches a high value close to 1/4 at θ = π/4 and r = 187.3 nm (see Section S3 of Supplement 1).

Fig. 3. (a) P₁ and (b) ${\hat{m}_3}$ of the scattered light at φ = 0 for (w_x, w_y, η) = (1, 1, 0), (c) P₁ of the scattered light at φ = 0 for (w_x, w_y, η) = (1, 1, 0.999).

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In the xOz plane (φ = 0), four elements of the polarization matrix, including R₁₃, R₂₃, R₃₁, and R₃₂, should be zero for unpolarized illumination and non-zero for partially polarized illumination. In this case, non-zero eigenvalues and the unitary matrix U of the polarization matrix change their value with the increase of correlation coefficient. In Fig. 3(c), the minimum value of P₁ is approximately the value of correlation coefficient. As P₁ denotes the weight of fully polarized component in the scattered light, the result is in line with expectation as a higher correlation coefficient surely leads to a higher weight of fully polarized state. If the incident beam is fully polarized, the value of P₁ will be constantly 1. Notably, it is found that ${\hat{\textbf{R}}_m}$ and ${\hat{m}_3}$ remain constant regardless of the variation of correlation coefficient (not plotted in Fig. 3) in the numerical calculation. As ${\hat{\textbf{R}}_m}$ represents the superposition of two uncorrelated orthonormal states, the variation of correlation coefficient only contributes to the change of ${\hat{\textbf{R}}_p}$.

The influences of partially polarization on polarization nonregularity and dimension are then calculated. Since the polarization nonregularity P_N is proportional to 1 – P₁ and ${\hat{m}_3}$, it is zero in the forward, backward, and side direction for unpolarized plane wave, as shown in Fig. 4(a). Theoretically speaking, a perfect nonregular polarization state of P_N = 1 requires P₁ = 0 (k₀r = 2.354) and ${\hat{m}_3}$= 1/4 (E_r = iE_θ and |E_r|² + |E_θ|² = |E_φ|²) that cannot be satisfied in Mie scattering. Nevertheless, numerical calculation shows that P_N reaches the maximum value of 0.928 at (r, θ) = (187.4 nm, 46.5°), suggesting that a nearly perfect nonregular polarization state appears in the near field. The polarization nonregularity drops intensively due to the attenuation of |E_r| with the propagation from near field to far field and finally approaches zero in the far field. Since the patterns of both P₁ and ${\hat{m}_3}$ change little for different correlation coefficients, Figs. 4(a), 4(c), and 4(e) exhibit similar distribution of P_N, despite its maximum value decreases with the increase of correlation coefficient. If the incident beam is fully polarized, P_N will be zero everywhere as the scattered light is also fully polarized. By comparison, the polarization dimension P_D shown in Figs. 4(b), 4(d), and 4(f) depends on the relative intensity of all the three components of light field. In the far field, P_D is bounded between 1 and 2 as E_r vanishes, where P_D = 1 denotes the linear polarization and P_D = 2 is reached for intensity-isotropic 2D light. In the near field, P_D > 2 for unpolarized incidence is observed, as shown in Fig. 4(b). As a large P_D requires that all the three components of electric field have considerable intensity, P_D reaches the maximum value of 2.7 at (r, θ) = (133.4 nm, 35.2°), which is close to that associated with the maximum value of P_N. With the increase of correlation coefficient, the maximum value of P_D gradually decreases and finally falls below 2 for fully polarized incidence, where P_D = 2 is reached for circularly polarized light.

Fig. 4. (a) Degree of nonregularity and (b) polarization dimension in the near field of scattered light for (w_x, w_y, η) = (1, 1, 0). (c) and (d) are for (w_x, w_y, η) = (1, 1, 0.5). (e) and (f) are for (w_x, w_y, η) = (1, 1, 0.999).

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It is well known that the spin is along the propagation direction for a paraxial beam, namely the spin is longitudinal. Here in Mie scattering, the spin exhibits complicated distribution in the near field due to nonzero radial components akin to the transverse spin in a tightly-focused beam. Figures 5(a)–5(c) shows the spin in the near field of scattered light for different correlation coefficients. For the incidence of unpolarized plane wave, the spin magnitude is axisymmetric about the z-axis and shows dependence on the polar angle in Fig. 5(a), where zero spin is observed along the z-axis and in the xOz plane. In Fig. 5(c), the spin still rotates about the z-axis and it is no longer zero in the xOy plane for the incidence of fully polarized plane wave. According to the characteristic decomposition, the total spin can be regarded as a vector sum of that included in ${\hat{\textbf{R}}_p}$ and ${\hat{\textbf{R}}_m}$. Therefore, the spin distribution depends on the weight of ${\hat{\textbf{R}}_p}$ and ${\hat{\textbf{R}}_m}$ when the incident light is not fully polarized. It is numerically found that the spin magnitude reaches the maximum value of 0.6 at (r, θ) = (114.5 nm, 33.3°) in the yOz plane for unpolarized illumination, where the polarization nonregularity is 0.4643. With the increase of correlation coefficient, the maximum value of spin increases from 0.6 at η = 0 to 0.7 at η = 0.5 to 1 at η = 1 monotonically.

Fig. 5. Spin in the near field of scattered light for (a) (w_x, w_y, η) = (1, 1, 0), (b) (w_x, w_y, η) = (1, 1, 0.5), and (c) (w_x, w_y, η) = (1, 1, 1). The arrow shows the direction and the color gives the magnitude. (d) Average direction cosine between the spin and radial direction at different distances from the sphere.

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Remarkably, the spin direction does not necessarily coincide with the radial direction in Mie scattering as that in paraxial beams, indicating that the spin may contain transverse component. To quantitatively examine the weights of transverse and longitudinal spin, we employed the absolute value of direction cosine between the spin and radial directions,

(20)$$\alpha = |{{\textbf{n}_s} \cdot {\textbf{n}_r}} |$$

as a metric, where n_s and n_r respectively denote the unit vectors of spin and radial direction. α = 0 and 1 correspond to pure transverse and longitudinal spin, respectively. Figure 5(d) gives the average value of absolute direction cosine at different radial distances from the nanosphere. The zero value of direction cosine suggests that the spin is transverse everywhere for unpolarized illumination (η = 0), which affects the energy flow based on the curl relationship [31]. Unlike unpolarized illumination, the spin contains both transverse and longitudinal components for nonzero η. Besides, the weight of transverse spin shows the dependency on the propagation distance. α decreases from 0.133 at r = 0.02 µm to 0.002 at r = 0.16 µm to 0.0015 at r = 1.8 µm for fully polarized illumination. It is because that the polar and azimuthal components of scattered light construct a 2D polarization ellipse with a major axis much longer than the minor axis in the region between r = 0.16 µm and 1.8 µm, in other words, the scattered light is quasi-linearly transverse to the propagation direction. In this case, the spin perpendicular to the electric field will subsequently be quasi-transverse. With further propagation into the far field, the transverse spin gradually approaches zero as the phase retardation between polar and azimuthal components increases and the radial component becomes negligible, which is in agreement with the phenomenon for paraxial beams. For partially polarized illumination, the dependency of α on the propagation distance is similar to that for fully polarized illumination. However, it cannot be simply regarded as a linear superposition of that for unpolarized and fully polarized illumination, as the spin is always longitudinal in the far field for partially polarized illumination.

5. Conclusion

In summary, we investigated polarization characteristics of light scattering by a dielectric nanosphere in the near field through the rigorous Mie scattering theory. Through characteristic decomposition of the polarization matrix, the polarization state is expressed in terms of an incoherent combination of pure and discriminating components, based on which the degree of nonregularity, polarization dimension, and electric-field spin are employed to characterize the light field. Numerical and analytical results show that for a partially polarized plane wave, the scattered light has a 3D polarization state in the near field, where the degree of nonregularity and polarization dimension generally increase with the decrease of correlation coefficient. Although the condition for perfect nonregular polarization cannot be satisfied, the degree of nonregularity reaches the maximum value of 0.928 in the limiting case of unpolarized incidence, where the state is a mixture of a nearly-circularly polarized state and a mutually orthogonal linearly polarized state. In addition, we show the spin is always transverse to the radial direction for unpolarized incidence, while the weight of transverse spin first rises and then falls with the propagation from the near field to far field for partially and fully polarized incidence. This work provides new insights into the complicated structure of 3D scattered light field, which are frequently encountered in the interaction between light and nanoparticle. It is expected that these findings could be useful in novel designs of nanoantenna, optical trap and sensing.

Funding

National Natural Science Foundation of China (52106162, 61605081).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Polarization characteristics and transverse spin of Mie scattering

Abstract

1. Introduction

2. Light scattering problem

3. Polarization characteristics

4. Results

5. Conclusion

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (5)

Equations (20)

Optics Express