Lateral shifts of linearly- and radially-polarized Bessel beams scattered by a nanosphere

Zhaolou Cao; Zhaolou Cao; Wei Liu; Wei Liu; Qi Sun; Qi Sun; Fenping Cui; Fenping Cui; Jinhua Li; Jinhua Li; Fenglin Xian; Fenglin Xian; Shixin Pei; Shixin Pei; Jia Liu

doi:10.1364/OE.447646

1. Introduction

It is known that an aperture-limited plane wave will experience lateral shift of a light beam from its expected geometrical optics path, namely Goos-Hänchen (GH) shifts [1], when it is totally reflected from the planar interface between two materials with different refractive indices. Originating from the angle-dependent complex reflectivity, the GH shift has attracted a lot of attention in micro/nano optics for their potential applications in sensing, as the reflectivity can be modulated by state-of-the-art metamaterials and metasurface. There have been numerous intriguing materials as reflection surfaces to investigate the GH shift in recent years, such as graphene-coated surface [2–4], weakly absorbing media [5,6], photonic crystals [7,8], and epsilon near-zero materials [9,10]. A giant GH shift up to several orders of wavelengths is observed in some cases, which makes it a feasible tool for sensing with an extremely high sensitivity. For example, GH shifts can be used to measure various quantities such as beam angle, refractive index, displacement, temperature, and film thickness in optical heterodyne sensors [11]. The phenomenon of GH shift can also be used for the characterization of the permeability and permittivity of the materials [12], optical differential operation and image edge detection [13], and in the development of near-field optical microscopy and lithography [14].

On the other hand, light scattering by particles is ubiquitous in our daily lives. The scattered light can be regarded as a fingerprint of the particle to some extent, which is utilized to retrieve the particle information in light scattering techniques. Despite that light scattering is an old topic, it has been continuously revisited from new perspectives. Many efforts have been devoted to investigating the influences of beam profile on the scattering, like Bessel beams [15], Airy beams [16], and Gaussian-Laguerre beams [17]. Generally speaking, these works focus on the distribution of scattered light or optical force, showing that it is possible to manipulate the scattered light through modulating the beam profile.

In 2014, J. Soni et al. [18] showed that a lateral beam shift, analog to the GH shift, exists when a plane wave is scattered by a nanosphere. Giant lateral shifts are observed due to the interference of neighboring localized plasmon resonance modes. Unlike traditional GH shifts at the planar interface, the lateral shift in Mie scattering originates from non-trivial radial components of scattered light in the far-field. Such phenomena need to be considered in the interpretation of nanometric displacement measurements via particle scattering [19–21]. However, it is still unclear how the lateral shift changes when beam profiles and polarization states are modulated.

In this paper, motivated by the above introduction, a numerical approach is developed to investigate the lateral shifts of linearly-polarized (LP) and radially-polarized (RP) Bessel beams, selected as two representatives of shaped vectorial beams, scattered by a nanosphere. This paper is organized as follow. In Section 2, we present detailed information of the numerical procedure. Results and discussions of the simulation are given in Section 3, while conclusions are drawn in Section 4.

2. Numerical procedure

Figure 1 shows the schematic of the problem to be investigated. A polarized shaped beam illuminates a sphere and generates non-uniformly distributed scattered light. The sphere with a radius of a₀ and refractive index of n₀ is assumed to be isotropic and homogenous. A global Cartesian coordinate system (x_G, y_G, z_G) is attached to the sphere with its origin located at the sphere center and its corresponding global spherical coordinate system is denoted as (r_G, θ_G, φ_G), where r, θ, and φ respectively denote the radial distance, polar angle, and azimuthal angle.

Fig. 1. Illustration of Bessel beam scattered by a nanosphere. P_r, P_θ, and P_φ respectively denote the radial, angular, and azimuthal components of Poynting vector of scattered light.

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Following the work in Ref. [18], the lateral shift determined by the localized beam’s centroid position is defined utilizing the Poynting vector, given by

(1)$${\vec{\Delta }_L} = \lim r\left( {\frac{{{P_\theta }\hat{\theta }}}{{|{{P_r}} |}}} \right), $$

where P_r and P_θ are respectively the radial and angular components of the Poynting vector, given by,

(2)$${\vec{P}_s} = {\vec{E}_s} \times {\vec{H}_s}, $$

where ${\vec{E}_s}\textrm{ = }({{E_r},{E_\theta },{E_\varphi }} )$ and ${\vec{H}_s}\textrm{ = }({{H_r},{H_\theta },{H_\varphi }} )$ are respectively the electric and magnetic fields of the scattered light, given by

(3)$$\begin{aligned} {{\vec{E}}_s} &= \sum\limits_{n = 1}^\infty {{E_n}({i{a_n}\vec{N}_{e1n}^{(3)} - {b_n}\vec{M}_{o1n}^{(3)}} )} ,\\ {{\vec{H}}_s} &= \frac{k}{{\omega \mu }}\sum\limits_{n = 1}^\infty {{E_n}({i{b_n}\vec{N}_{o1n}^{(3)} + {a_n}\vec{M}_{e1n}^{(3)}} )} , \end{aligned}. $$

For the sake of brevity, detailed expressions of variables in Eq. (3) are not given here and can be found in Eq. (4.50) in Ref. [22].

To date, various methods have been developed to solve the particle scattering problem, like T-Matrix method, angular spectrum theory, and generalized Lorenz-Mie theory. The angular spectrum theory, known as the basis for Fourier Optics [23], is employed for its easy implementation and high computational efficiency. Be decomposing the shaped beam into a series of plane waves, the scattered light of shaped beam can be easily obtained by superposing that of plane waves weighted by coefficients related to the beam profile. Several works have already applied the method in solving the scattering problem [24,25]. Here, a numerical procedure is also developed based on the angular spectrum theory. According to the angular spectrum theory, a scalar shaped beam with an arbitrary shape can be expressed as

(4)$${E_i} = {E_0}\int {F({{n_x},{n_y}} )exp [{i{k_0}({{n_x}{x_i} + {n_y}{y_i} + {n_z}{z_i}} )} ]{d^2}K}, $$

where E₀ is a constant controlling the magnitude, F(n_x, n_y) denotes the angular spectrum of the incident beam, k₀ = 2π/λ is the wave number, λ is the wavelength, n_x, n_y and n_z are the direction cosines of plane wave, subjected to n_x² + n_y² + n_z² = 1. As Mie scattering is highly sensitive to the polarization, the scalar beam has to be vectorized to take account of the vectorial nature of electromagnetic field. Here, linear polarization states with directions along the x-axis and z-axis are assumed for the scalar shaped beam for the sake of simplicity. As the polarization state can never be perpendicular to the propagation direction of every decomposed plane wave for a two-dimensional beam, the assumed polarization is then tailored for propagating in the far field. For a plane wave with a propagation direction given by

(5)$$\vec{k} = {k_0}({{n_x}{{\vec{e}}_x} + {n_y}{{\vec{e}}_y} + {n_z}{{\vec{e}}_z}} ), $$

and polarization direction given by $\vec{f}$, its vectorial amplitude is given by

(6)$$\vec{E}({{n_x},{n_y}} )= {E_0}F({{n_x},{n_y}} ){\vec{f}_ \bot }({{n_x},{n_y}} ), $$

where ${\vec{e}_x}$, ${\vec{e}_y}$, and ${\vec{e}_z}$ are respectively the unit vector along the x-, y-, and z-axis,

(7)$${\vec{f}_ \bot }({{f_x},{f_y}} )= \vec{f} - \frac{{\vec{k}}}{{{k_0}}}\left( {\frac{{\vec{k}}}{{{k_0}}} \cdot \vec{f}} \right), $$

In Eq. (7), the vectorization removes the polarization component along the propagation so that the constructed polarization direction is always perpendicular to the propagation direction. For convenience, axisymmetrical shaped beams with $\vec{f}$ of ${\vec{e}_x}$ and ${\vec{e}_z}$ are respectively referred as linearly-polarized (LP) and radially-polarized (RP) beams in the following text.

The scattered field of a shaped vectorial beam can then be obtained by linearly superposing that of plane waves, given by

(8)$${\vec{E}_{s,shape}}({{r_G},{\theta_G},{\varphi_G}} )= \int_{{n_x},{n_y}} {{{\vec{E}}_{s,p}}({{n_x},{n_y};{r_G},{\theta_G},{\varphi_G}} )F({{n_x},{n_y}} )d{n_x}d{n_y}}, $$

where the subscripts shape and p respectively denote the shaped beam and plane wave,

(9)$${\vec{E}_{s,p}}({{n_x},{n_y};{r_G},{\theta_G},{\varphi_G}} )= R({{n_x},{n_y};{\theta_L},{\varphi_L}} ){\vec{E}_{s,p}}({{r_L},{\theta_L},{\varphi_L}} ), $$

where R(n_x, n_y; θ_L, φ_L) is a rotating matrix transforming the electric field at the position of (r_L, θ_L, φ_L) in the localized spherical coordinate system to that at the position of (r_G, θ_G, φ_G) in the global spherical coordinate system. The associated Cartesian systems of the two spherical coordinate systems are transformed as

(10)$$\begin{aligned} {{\vec{e}}_{zL}} &= {n_x}{{\vec{e}}_{xG}} + {n_y}{{\vec{e}}_{yG}} + \sqrt {1 - n_x^2 - n_y^2} {{\vec{e}}_{zG}}\\ {{\vec{e}}_{xL}} &= \left\{ {\begin{array}{*{20}{c}} {\frac{{{{\vec{e}}_{xG}} - ({{{\vec{e}}_{xG}} \cdot {{\vec{e}}_{zL}}} ){{\vec{e}}_{zL}}}}{{|{{{\vec{e}}_{xG}} - ({{{\vec{e}}_{xG}} \cdot {{\vec{e}}_{zL}}} ){{\vec{e}}_{zL}}} |}}\;\;\;\;for\;LP\;beams}\\ {\frac{{{{\vec{e}}_{zG}} - ({{{\vec{e}}_{zG}} \cdot {{\vec{e}}_{zL}}} ){{\vec{e}}_{zL}}}}{{|{{{\vec{e}}_{zG}} - ({{{\vec{e}}_{zG}} \cdot {{\vec{e}}_{zL}}} ){{\vec{e}}_{zL}}} |}}\;\;\;\;for\;RP\;beams} \end{array}} \right.\\ {{\vec{e}}_{yL}} &= {{\vec{e}}_{zL}} \times {{\vec{e}}_{xL}} \end{aligned}, $$

where the subscripts L and G respectively denote the localized and global coordinate systems.

Bessel beams are known for their non-diffractive propagation. They can be regarded to be formed out of a conical interference of an infinite number of plane waves that cross the optical axis at the same angle, referred as the conical angle of θ_Axis. Each propagating wave undergoes the same phase shift over a distance. This decomposition of the Bessel beam into plane waves manifests itself in the angular spectrum of the beam as a ring, given by

(11)$${F_{Bessel}}({{n_x},{n_y}} )= \frac{1}{{2\pi \sin {\theta _{Axis}}}}\delta ({n_x^2 + n_y^2 - {{\sin }^2}{\theta_{Axis}}} ), $$

where θ_Axis determines the main lobe radius.

The constructive interference on the optical axis produces an intense central spot surrounded by several cylindrically symmetric lobes with lower intensity. Note that if the scatterer is not located at the center of the beam, an additional phase term of exp[ik₀(n_xx_d+n_yy_d)] should be added in Eq. (11), where x_d and y_d respectively denote the scatterer shift from the beam center in the x and y direction. For a specific scatterer, its scattered light distribution can be solved and saved in the memory before performing the integration in Eq. (8). By this way, there is no need to repeatedly solve the Mie scattering when the scatterer is illuminated by different beams. Therefore, the proposed method has a higher computational efficiency than the generalized Lorenz-Mie theory (GLMT) [26] that needs to solve beam shape coefficients for different beam profiles. A step-by-step description of the simulation is given below.

Step 1. Represent a scalar shaped beam with plane waves with weighted amplitude by the angular spectrum theory. Note that the weighted amplitude may have analytical expressions for beams that have analytical expressions, like Gaussian beam, Bessel beam and Airy beam.

Step 2. Vectorize each polarized plane wave component and update the polarization direction and amplitude.

Step 3. Calculate the classical plane wave scattering by a sphere to construct a database consisting of the electromagnetic fields at different positions.

Step 4. For a given position in the global coordinate system, referred as P_g for convenience, it is transformed into the localized coordinate system for each plane wave component.

Step 5. The electromagnetic field at a specific position in the localized coordinate system is evaluated by linear interpolation utilizing the data in Step 2 so that Mie scattering problem does not need to be repeatedly solved. Since the scattered light of nanospheres varies smoothly in the space, the interpolation will not introduce observable numerical error in the results.

Step 6. Superpose the scattered light of each plane wave to get that of the shaped beam at the position of P_g.

Step 7. Repeat Step 4-6 for each position in the global coordinate system to obtain the light field in the entire space.

3. Results

In this section, a series of numerical calculations were performed to investigate the lateral shift of LP and RP Bessel beams during the Mie scattering based on the proposed numerical approach. The spatial distribution of LP and RP Bessel beam in the xOz plane is shown in Fig. 2, where θ_Axis is 0.8 rad for both LP and RP Bessel beams. For the LP Bessel beam, the x-component of electric field appears as a Bessel function in the x-direction, while the z-component exhibits as a hollow beam on the z-axis. As the maximum intensity of x-component is much higher than that of z-component, the wavefront of a LP Bessel beam can be regarded to be similar to that of a LP plane wave near the z-axis. In Fig. 2(c), the full-width-at-half-maxima (FWHM) radius of the mainlobe is approximately 110 nm. Considering it is much larger than a nanosphere, the scattered light of a LP Bessel beam should have similar distribution to that of a LP plane wave. In contrast to the LP Bessel beam, it is the z-component that appears as a Bessel beam for the RP Bessel beam. As the x-component is a hollow beam, an on-axis nanosphere seems to be exposed to a plane wave with a polarization state along the propagation direction. Despite light scattering of a LP plane wave has been extensively studied in the literatures, few works considered such a plane wave that is incapable of propagating in the far field. Here, the RP Bessel beam offers an opportunity to study the scattering of such a plane wave by a nanosphere. Figure 2(g) shows the dependence of the FWHM mainlobe radii on θ_Axis for |E_x|² of the LP Bessel beam and |E_z|² of the RP Bessel beam. The RP Bessel beam always has a slightly smaller FWHM radius than the LP Bessel beam. With the increase of θ_Axis, the FWHM radius gradually decreases and reaches a limit due to the diffraction limit.

Fig. 2. The intensity of (a) x- and (b) z- components of electric field in the xOz plane. (c) Normalized intensity distribution at z = 0 μm. (a), (b), and (c) are for the x-polarized Bessel beam. (d), (e), and (f) are the same as (a), (b), and (c), but for the RP Bessel beam. Parameters are: λ = 0.4 μm, θ_Axis = 0.8 rad. (g) Dependence of the FWHM mainlobe radii on θ_Axis for |E_x|² of the LP Bessel beam and |E_z|² of the RP Bessel beam.

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A metal nanosphere with a refractive index n_p of 0.05 + 2.1i (Ag at λ = 0.4 μm) is employed in the following simulation as lateral shifts vanish for small dielectric Rayleigh scatterers (a₀ << λ). Comparison of lateral shifts among a LP plane wave, a LP Bessel beam and a RP Bessel beam in the far field is given in Fig. 3. The pattern is symmetry to the axis of φ = 0° due to the symmetry of light scattering to the xOz plane. A maximum lateral shift of 2.5λ is observed at [θ, φ] = [93°, 0°], while the minimum lateral shift is -2.5λ at [θ, φ] = [85°, 0°]. The lateral shift is negligible outside the xOz plane for a LP plane wave. Note that positions of maximum and minimum lateral shift are sensitive to illumination and particle parameters, and it is difficult to accurately predict the lateral shift by analytical expressions. According to Eq. (1), the lateral shift is related to the value of P_r = E_θH_φ - E_φH_θ and P_θ= E_φH_r – E_rH_φ. As E_φ is zero in the xOz plane, a near-zero E_θ will lead to a near-zero P_r and subsequently a giant lateral shift. The inset giving the angular dependence of |E_θ| for a₁ dipolar and a₂ quadrupolar modes shows that the a₁ mode has a zero value of |E_θ| at θ = 90°. Therefore, the lateral shift will exhibit local peaks near θ = 90° if the overall magnitude of |E_θ| for the a₂ mode is much smaller than that for the a₁ mode. Since P_θ changes the sign in this region, the sign of lateral shift goes from negative to positive. Note that the sign change does not necessarily happen for particles with different parameters.

Fig. 3. Lateral shifts for (a) a LP plane wave, (b) a LP Bessel beam, and (b) a RP Bessel beam scattered by an on-axis nanosphere. Light intensity expressed for the scattering of (d) the LP plane wave, (e) the LP Bessel beam, and (f) the RP Bessel beam with arrows representing the direction of lateral shift. Parameters are:a₀ = 50 nm, λ = 0.4 μm, θ_Axis = 0.8 rad. Both LP plane wave and LP Bessel beam are x-polarized. θ and φ respectively denote the polar and azimuthal angles in the spherical coordinate system.

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Figure 3(b) gives the lateral shift of a LP Bessel beam. The similar pattern to that in Fig. 3(a) demonstrates that the LP Bessel beam does not significantly reshape the lateral shift. However, the maximum and minimum lateral shifts respectively increase to 3.8λ and decrease to -3.8λ. Except for the region near maximum and minimum lateral shifts, other parts still show little lateral shift. By comparison, lateral shifts of RP Bessel beams show a completely different pattern in Fig. 3(c). The lateral shift does not have any azimuthal dependence but is constant along the φ direction because of the axisymmetry about the z-axis of a RP Bessel beam. Note that the data for θ = 0°, 1°, 2°, 178°, 179°, and 180° is not plotted in Fig. 3(c) as there are two singularities of lateral shifts at θ = 0° and 180°. It is because that the angular and azimuthal components of electric field should be zero at θ = 0° and 180° due to the axisymmetry, which lead to a zero radial component of Poynting vector. In other parts, the lateral shift in the θ direction varies smoothly and monotonously in Fig. 3(c), which also differs from that in Figs. 3(a) and 3(b), indicating that light scattering of a plane wave with a polarization state along the propagation direction is quite different from that of a conventional LP plane wave in essence. Scattered light intensity evaluated by I = |E_r|² + |E_θ|² + |E_φ|² for the three beams is shown in Figs. 3(d), 3(e), and 3(f), where superposed arrows illustrate the direction and magnitude of lateral shift. It can be found that the lateral shift generally points from low intensity to high intensity.

Since the main lobe radius increases with the decrease of θ_Axis, the Bessel beam scattering shall reduce to the plane wave scattering as θ_Axis approaches to zero. Figure 4 gives the influences of θ_Axis, where lateral shifts generally have the same trend for different θ_Axis in the xOz and yOz planes. The magnitude of maximum and minimum lateral shifts gradually increases with the increase of θ_Axis in the xOz plane. However, the curve is not scaled up as a whole in the ordinates, but keep unchanged in the area far from positions associated with the maximum and minimum value. In Fig. 4(b), large lateral shifts are observed near φ = 0° and ±180° in the yOz plane, which also have positive correlation with θ_Axis. As the magnitude of lateral shift quickly drops to nearly zero in other regions, the following analysis focuses on the xOz plane solely.

Fig. 4. Influences of θ_Axis on lateral shifts in the (a) xOz plane (φ = 0 rad) and (b) yOz plane (θ = 0 rad) for the LP Bessel beam. (c) The maximum value of Δ_L/λ as a function of θ_Axis. (d) Influences of θ_Axis on the lateral shift in the xOz plane (φ = 0 rad) for the RP Bessel beam. Parameters are: a₀ = 50 nm, λ = 0.4 μm. The inset shows the variation of |E_θ| as a function of θ for the two lowest modes: a₁ dipolar mode and a₂ quadrupolar mode.

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For a LP Bessel beam, the beam incidenting on a nanosphere can be regarded as a LP plane wave with an equivalent wave number of k₀cosθ_Axis and wavelength of λ/cosθ_Axis. Since the lateral shift is divided by the original wavelength in the ordinates, it can be qualitatively estimated to be increased by 1/cosθ_Axis times. Figure 4(c) plots the comparison between numerical simulation and qualitative estimation of the maximum value of Δ_L/λ. Despite that the numerical solution is slightly larger than the qualitative estimation, the qualitative estimation is still capable of revealing the trend, demonstrating that the explanation of equivalent wavelength can be helpful of qualitatively understanding how the lateral shift of a LP Bessel beam changes compared with that of a plane wave.

Unlike the LP Bessel beam, lateral shifts of RP Bessel beams show little dependence on θ_Axis. In Fig. 4(d), the curves for θ_Axis = 0.4 rad, 0.8 rad, and 1.2 rad nearly overlap each other. This result suggests that light scattering by a plane wave with a polarization state along the propagation direction has a weak correlation with the equivalent wavelength that has also been increased by 1/cosθ_Axis times for a RP Bessel beam.

Due to non-uniform intensity profiles, discrepancies of light scattering between Bessel beam and plane wave should increase with increasing the nanosphere radius. Figure 5 gives results for a nanosphere with a radius of 100 nm. In Fig. 5(a), a giant lateral shift is observed at θ = 154° for the LP plane wave scattering. It is because that the overall magnitude of a₂ quadrupolar mode increases due to surface plasmon resonance and low localized plasmon modes experience destructive interference at the position, exhibiting a near-zero |E_θ| as shown in Fig. 5(b). Detailed descriptions of the interference among plasmon modes can be found in Ref. [18]. The near-zero radial component of Poynting vector then generates a giant lateral shift, demonstrating that the interference among different plasmon modes will produce giant lateral shifts at positions other than θ near 90°. The giant lateral shift vanishes for the LP Bessel beam scattering as the superposition of different plane waves eliminates the destructive interference. In other regions, electric fields and lateral shifts for the LP plane wave and Bessel beam show similar trends despite of shifts of localized maxima and minima. Figure 5(c) gives the comparison of lateral shifts between a₀ = 50 nm and 100 nm for a RP Bessel beam. The two curves almost overlap each other inside θ ∈ [0°, 30°] and [150°, 180°]. As shown in Fig. 5(d), a localized minimum of |E_θ| is observed at θ = 109° for a₀ = 100 nm, which also originates from the destructive interference among different plasmon modes and leads to a weak oscillation of lateral shifts in the region in Fig. 5(c).

Fig. 5. Comparison of (a) lateral shifts and (b) electric field between a LP Bessel beam and plane wave for a nanosphere with a₀ of 100 nm. Comparison of (c) lateral shifts and (d) electric field of a RP Bessel beam between a₀ = 50 nm and 100 nm. Parameters are: λ = 0.4 μm, θ_Axis = 0.8 rad.

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Unlike plane wave scattering, Bessel beam scattering is also influenced by the nanosphere position. We simulated the evolution of lateral shifts when the nanosphere moves away from the z-axis in the x- and y-direction, results of which are given in Fig. 6. For both LP and RP Bessel beams, lateral shifts are more sensitive to the x-shift than the y-shift. A shift of 100 nm in the x-direction leads to a great change of lateral shifts in Fig. 6(a), while lateral shifts change little when the y-shift is less than 400 nm. As the nanosphere moves out of the main lobe, the magnitude of lateral shifts gradually decreases for the LP Bessel beam in Figs. 6(a) and 6(b). By comparison, RP Bessel beam shows a simlar trend of lateral shift to that for LP Bessel beams in Fig. 4(a) when the x-shift is 200 nm. We believe the reason is that the RP Bessel beam can be approximated to be a plane wave with a x-polarization at (x, y) = (200 μm, 0 μm), where E_z approaches to zero and E_x dominates the electric field as shown in Fig. 2. In conformity with this explanation, lateral shifts gradually vanishes with the increase of y₀ in Fig. 6(d). It is because that the lateral shifts for LP plane waves with x- and y-polarization exhibit symmetry with respect to the azimuthal angle. lateral shifts in the xOz plane for non-zero y-shifts should be equivalent to that in the yOz plane for non-zero x-shifts, which is nearly zero outside the xOz plane as shown in Fig. 3(a).

Fig. 6. Influences of the (a) x-shift and (b) y-shift of the nanosphere on lateral shifts for the LP Bessel beam. (c) and (d) are the same as (a) and (b), but for the RP Bessel beam. Parameters are: a₀ = 50 nm, λ = 0.4 μm, θ_Axis = 0.8 rad.

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4. Conclusion

In conclusion, a numerical approach is developed to solve light scattering of an arbitrary shaped beam by a nanosphere with a high computational efficiency based on the angular spectrum theory. Non-diffractive Bessel beams are decomposed as an ensemble of plane waves with different propagation directions and polarization states. Scattered light of each plane wave is then superposed to get that of LP and RP Bessel beams after coordinate transformation. Numerical results show that a LP Bessel beam can be regarded as a plane wave with the same polarization state but an equivalent longer wavelength for on-axis scattering, while a RP Bessel beam can be regarded as a plane wave with a polarization state along the propagation direction. Besides, it is shown that lateral shifts defined based on the Poynting vector are closely related to the angular component of electric field. A small angular component of electric field usually leads to a giant lateral shift in the xOz plane. As nanospheres can be used as probes in nanometric displacement measurement, the findings will deepen our understandings of lateral shifts in light scattering of vectorial non-diffractive beams.

Funding

Jiangsu Provincial Key Research and Development Program (BE2020006-2); Natural Science Foundation of Jiangsu Province (BK20150929, BK20180784, BK20180799); National Natural Science Foundation of China (11605090, 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.

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Lateral shifts of linearly- and radially-polarized Bessel beams scattered by a nanosphere

Abstract

1. Introduction

2. Numerical procedure

3. Results

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (11)

Optics Express