Unusual optical phenomena inside and near a rotating sphere: the photonic hook and resonance

Huan Tang; Zhuoyuan Shi; Yuan Zhang; Renxian Li; Bing Wei; Shuhong Gong; Igor V. Minin; Oleg V. Minin

doi:10.1364/OE.518794

1. Introduction

Since the motion electrodynamics of rigid bodies was first proposed by Einstein when elaborating the principle of relativity [1], although this was only a hypothesis, extensive research has been carried out [2,3]. The studies around rotational scattering bodies have a research hotspot, which mainly converges in the electromagnetic field. The wave vector inside a two-dimensional rotating target illuminated by the electromagnetic field when it rotates at a low speed is derived by C.T.Tai [4]. After that, the rotating scattering characteristics of ground glass [5], electron-plasma column [6], dielectric cylinder [7,8], circular conducting cylinder [9,10], and coated metamaterial cylinders [11] are studied and discussed gradually. In the process, a relativistic solution to the problem of scattering by a rotating dielectric column is attained, which makes rotating targets no longer limited by low-speed conditions [12]. With the further deepening of the rotational scattering theory, the scattering characteristic of three-dimensional rotating targets is concerned. Both the rotating dielectric [13] and conducting [14] spheres are explored. Gradually, the mutual subject of beam and rotating targets has attracted attention and the characteristic has been revealed with the development of optical research. Similar to the Magnus effect in hydrodynamics [15], the transverse component of the optical radiation force appears when a plane wave exerts on rotating particles, which is named as optical Magnus effect. Plenty of research results [16–20] have been generated based on it, which has been proven to apply to the optical tweezers [21] for particle manipulation [22,23] and topological spin transport of photons [24]. Although current research on the optical Magnus effect has not been widely applied in engineering similar to the Sagnac effect [25], it has already played a significant role in fiber optic gyroscopes [26] and Global Positioning Systems [27], its application potential is still enormous.

The photonic nanojet (PNJ) [28] is a weakly diffracted non-resonant beam with much higher light intensity than an incident wave in the shadow-side surfaces of micron-scale particles. It occupies a significant position in particle manipulation because of the weak sub-diffracting waist and a shallow divergence angle. It has been applied to the field of optical data storage [29], super-resolution imaging [30], optical manipulation [31,32], and nanoparticle detection [33]. However, the manipulation of the off-axis particles in three-dimensional space is relatively difficult for PNJ because the transmission direction always follows the axial direction. Therefore, a curved photonic nanojet named PH [34,35] is proposed by Minins to achieve simpler 3D manipulation of particles [36]. Extensive research [37–39] has been carried out around it, and it will shine in other fields including THz, acoustics, and plasmonics [40]. Normally, the structural asymmetry of particles, beams, or environment is the generation mechanism of PHs. Therefore, current mainstream research is focused on designing particle or beam structures to achieve PHs [41]. Recently, research has proposed a novel method to design PHs utilizing the rotation of particles without changing the structure of targets and incident beam based on the optical Magnus effect [42]. In this study, a new method for designing optical resonance cavities was also proposed, which utilizes rotation. Unfortunately, the article did not conduct a specific study on this phenomenon.

The WGM resonance [43] was proposed by Gustav Mie, who explained the mechanism of a spherical cavity theoretically. It was first discovered by Lord Rayleigh when he explored acoustic wave propagation phenomena appearing in St Paul’s Cathedral [44,45]. In 1939, the electromagnetic resonance phenomena were first analyzed in a spherical resonance cavity by Richtmyer [46], which theoretically predicted that spherical particles can maintain resonance modes with high-quality factors. Based on the above theory, the first WGM microspherical laser resonator was designed [47] in 1961. With the development of laser spectroscopy technology, research of the resonance micro-cavity has entered a fast lane [48–51]. Because of the high-quality factors, integrability, and easy coupling characteristics of the WGM resonance micro-cavity, it has been applied to the optical filter [52], optical sensor [53,54], and biomedical molecules [55]. In recent years, a super-resonance cavity [56–59] based on the high-order Fano resonance to generate giant magnetic and electric fields has been proposed (Note that this effect is valid for spherical particle only instead of WGM). It has promising application potential for the generation of photonic magnetic nanojet [60] and white light super-lens nanoscopy [61]. Above all things, it also might be a method to optimize the WGM resonance cavities [62].

Recently, it has been verified that both the PNJ and resonance can coexist simultaneously [63], and the resonance plays a significant role in the PNJ [64], which increases the effective strength of PNJ and reduces its full width at half maximum (FWHM) [65]. In addition, studies have found that the transverse structure of PNJ becomes complex (with a segmented structure) and its focus moves closer to particle surface [60,66,67] when resonance scattering occurs. What’s more, the intrinsic mechanism of the resonance scattering impacting PNJs has been revealed [68]. Although extensive research on the interaction between resonance and PNJ is being conducted, the impact of resonance scattering on PNJs is currently blank. Whether resonance will affect the magnitude of PHs curvature remains a mystery. Therefore, it is meaningful to explore the characteristics of PHs under resonance conditions, which guides 3D particle manipulation.

Therefore, the interaction between a rotating dielectric sphere and a polarized plane wave is explored in this manuscript. Both the PH and resonance caused by rotation are analyzed. The impact of resonance exerting on the PH is discussed. Besides, the influence of particle rotational velocity on the PH and resonance is emphasized. Although the above characteristics only occur when the particle accelerates to a higher speed, there is research [69,70] that provides a practical basis for the implementation of super-fast spin of particles under experimental conditions. Therefore, the idea in this paper provides theoretical guidance for the design of PHs and resonance cavities, which are meaningful in particle manipulation, optical sensors, and spectroscopy.

The overall layout of this article is as follows. In section 2, the scattering theory of a polarized plane wave exerts on a dielectric sphere rotating at a stable angular velocity is derived utilizing the multipole expansion method (MEM) [71] using vector spherical wave functions (VSWFs) [72] based on the "instantaneous rest-frame" hypothesis and Minkowski’s theory [2,4]. In section 3, the numerical results on the PH and resonance generated by rotation are calculated, and a detailed investigation into the other optical properties caused by rotation is conducted. In section 4, the innovation and conclusion of this article are summarized.

2. Scattering by a rotating dielectric sphere

As shown in Fig. 1(a), a dielectric sphere with radius $a$ located in a spherical coordinate system $(r, \theta, \varphi )$ whose center is $O$ spinning with a stable angular velocity $\Omega$ around $z$-axis is illuminated by a polarized plane wave with incident direction ${{\bf {u}}_i}$, surrounded by a non-magnetic, lossless and homogeneous medium. In Fig. 1(b) which depicts the wave vector in the laboratory coordinate system, both the transmission direction and polarization states of the incident waves are defined by a coordinate system $(\alpha, \beta, k)$ and orientation angle ${\alpha _{pol}}$. Where, ${i}_\alpha$, ${i}_\beta$, and ${i}_k$ are the components of unit vector $\bf {i}$ in the $\alpha$, $\beta$, and $k$ directions, respectively.

Fig. 1. Panel (a) displays the rotating dielectric sphere, with center $O$ rotating around the $z$-axis, illuminated by a plane wave propagating along an arbitrary direction. Panel (b) is the diagrammatic sketch describing the wave vector and polarization states of the incident wave.

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Based on the Maxwell equations of rotating material as follows [73,74]

(1)$$\begin{aligned} &{\nabla \times {\bf{E}} ={-} \frac{\partial }{{\partial t}}({\mu _1}{\bf{H}} - {\bf{\Lambda }} \times {\bf{E}})}\\ &{\nabla \times {\bf{H}} = \sigma ({\bf{E}} + {\mu _1}{\bf{v}} \times {\bf{H}}) + \frac{\partial }{{\partial t}}({\varepsilon _1}{\bf{E}} + {\bf{\Lambda }} \times {\bf{H}})} \end{aligned}$$

with ${\bf {\Lambda }} = \left ( {{\varepsilon _1}{\mu _1} - {\varepsilon _2}{\mu _2}} \right ){\bf {v}}$, and ${\bf {v}} = \Omega r\sin \theta {\bf {\varphi } }$. ${\bf {\varphi } }$ indicates the dielectric sphere rotating in ${\bf {\varphi } }$ direction, where $\varepsilon _1$ and ${\bf {v}}$ are the permittivity and spinning velocity of the particle. Considering a non-magnetic and uncharged particle, conductivity $\sigma = 0$, the permeability of sphere and medium is ${\mu _1} = {\mu _2}= {\mu _0}$, and ${\mu _0}$ is the permeability of free space. Therefore, Eq. (1) is simplified as follows

(2)$$\begin{aligned} &{(\nabla + {\rm{j}}\omega {\bf{\Lambda }}) \times {\bf{E}} = {\rm{j}}\omega {\mu _0}{\bf{H}}}\\ &{\left( {\nabla + {\rm{j}}\omega {\bf{\Lambda }}} \right) \times {\bf{H}} ={-} {\rm{j}}\omega {\varepsilon _1}{\bf{E}}} \end{aligned}$$

In the system where a rotating sphere is illuminated by a polarized plane wave, the electric field inside a rotating sphere satisfies the following equation after eliminating ${\bf {H}}$ using Eq. (2)

(3)$${\nabla ^2}{\bf{E}} + 2{\rm{j}}\omega ({\bf{\Lambda }} \cdot \nabla ){\bf{E}} + {\omega ^2}{\varepsilon _1}{\mu _0}{\bf{E}} = 0$$

The arbitrary component of electric field is expressed as $E = R\left ( r \right )\Theta \left ( \theta \right )\Phi \left ( \varphi \right )$ in a spherical coordinate system. Substitute it into Eq. (3) as follows

(4)$$\begin{aligned} \frac{{\Theta \Phi }}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{{\partial R}}{{\partial r}}} \right) + \frac{{R\Phi }}{{{r^2}\sin \theta }}\frac{\partial }{{\partial \theta }}\left( {\sin \theta \frac{{\partial \Theta }}{{\partial \theta }}} \right) + \frac{{{\partial ^2}\Phi }}{{\partial {\varphi ^2}}}\frac{{R\Theta }}{{{r^2}{{\sin }^2}\theta }}\\ + 2i\omega {\mu _{\rm{0}}}\left( {{\varepsilon _1} - {\varepsilon _2}} \right)\Omega r\sin \theta \frac{{R\Theta }}{{r\sin \theta }}\frac{{\partial \Phi }}{{\partial \varphi }} + {\omega ^2}{\varepsilon _1}{\mu _0}R\Theta \Phi = 0 \end{aligned}$$

Based on the method of separating variables, simplify it and only consider the radial component of the electric field, Eq. (4) is expressed as follows

(5)$$\frac{1}{R}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{{\partial R}}{{\partial r}}} \right) - n\left( {n + 1} \right) + \left[ {{\omega ^2}{\varepsilon _1}{\mu _0} - 2m\omega {\mu _{\rm{0}}}\left( {{\varepsilon _1} - {\varepsilon _{\rm{2}}}} \right)\Omega } \right]{r^2} = 0$$

Equation (5) satisfies the Bessel equation in a spherical coordinate system and is given as

(6)$$\frac{1}{R}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{{\partial R}}{{\partial r}}} \right) - n\left( {n + 1} \right) + {\left( {{\kappa _m}r} \right)^2} = 0$$

where ${\kappa _m}$ is the wave number inside a spinning dielectric sphere, which is derived by comparing the Eqs. (5)–(6) as

(7)$${\kappa _m} = \sqrt {m_1^2{k^2} - 2m\gamma \left( {m_1^2 - m_2^2} \right){k \mathord{\left/ {\vphantom {k a}} \right. } a}}$$

where $m_1$ and $m_2$ are the relative refractive index of a dielectric sphere and medium surrounding the sphere. $k = {{2\pi } \mathord {\left /{\vphantom {{2\pi } \lambda }} \right.} \lambda }$ is the wave number of an incident plane wave in a vacuum with wavelength $\lambda$. $\gamma = {{\Omega a} \mathord {\left /{\vphantom {{\Omega a} c}} \right.} c}$ is the non-reciprocal rotating dimensionless parameter, and $\gamma \ll 1$ based on the "instantaneous rest-frame" hypothesis. $c$ is the speed of light in a vacuum.

A time-harmonic factor ${e^{ - \rm {j}\omega t}}$ is chosen in this derivation process, thus the expressions of electric fields are defined in the form of the multipole expansion [71] and vector spherical wave functions (VSWFs) [72] based on the Mie theory [75] reads as follows

(8)$${{\bf{E}}^{inc}}(r) = \sum_{n = 1}^\infty {\sum_{m ={-} n}^n {{E_0}\left[ {{A_{mn}}{\bf{M}}_{mn}^{(1)}\left( {{k_{{m_2}}}r} \right) + {B_{mn}}{\bf{N}}_{mn}^{(1)}\left( {{k_{{m_2}}}r} \right)} \right]} }$$

(9)$${{\bf{E}}^{sca}}(r) ={-} \sum_{n = 1}^\infty {\sum_{m -{-} n}^n {{E_0}\left[ {{a_{mn}}{\bf{M}}_{mn}^{(3)}\left( {{k_{{m_2}}}r} \right) + {b_{mn}}{\bf{N}}_{mn}^{(3)}\left( {{k_{{m_2}}}r} \right)} \right]} }$$

(10)$${{\bf{E}}^{{\rm{int}}}}(r) = \sum_{n = 1}^\infty {\sum_{m ={-} n}^n {{E_0}\left[ {{c_{mn}}{\bf{M}}_{mn}^{(1)}({\kappa _m}r) + {d_{mn}}{\bf{N}}_{mn}^{(1)}({\kappa _m}r)} \right]} }$$

Where ${{\bf {M}}_{mn}^{(\cdots )}}$ and ${{\bf {N}}_{mn}^{(\cdots )}}$ are the VSWFs, whose expressions are given as follows

(11)$${\bf{M}}_{mn}^{(1,3)}(\rho) = \frac{1}{{\sqrt {2n\left( {n + 1} \right)} }}\sqrt {\frac{{2n + 1}}{2} \cdot \frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} Z_n^{(1,3)}(\rho)\left[ {{\rm{j}}m\tilde \pi _n^{|m|}(\theta ){{\bf{e}}_\theta } - \tilde \tau _n^{|m|}(\theta ){{\bf{e}}_\varphi }} \right]{{\rm{e}}^{{\rm{j}}m\varphi }}$$

(12)$$\begin{aligned} &{\bf{N}}_{mn}^{(1,3)}(\rho) = \frac{1}{{\sqrt {2n\left( {n + 1} \right)} }}\sqrt {\frac{{2n + 1}}{2} \cdot \frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}}\\ &\times \left\{ {n(n + 1)\frac{{Z_n^{(1,3)}(\rho)}}{{\rho}}\tilde P_n^{|m|}(\cos \theta ){{\bf{e}}_r}} \right.\left. { + \frac{{{{\left[ {\rho Z_n^{(1,3)}(\rho)} \right]}^\prime }}}{{\rho}}\left[ {\tilde \tau _n^{|m|}(\theta ){{\bf{e}}_\theta } + {\rm{j}}m\tilde \pi _n^{|m|}(\theta ){{\bf{e}}_\varphi }} \right]} \right\}{{\rm{e}}^{{\rm{j}}m\varphi }} \end{aligned}$$

$Z_n^{(\cdots )}(\rho )$ corresponds to the spherical Bessel and Hankel functions of the first kind when the superscript is 1 and 3, respectively. The expression of $\rho$ inside and outside the particle is $k_{m_2}r$ and $\kappa _m r$ with $r = \sqrt {{x^2} + {y^2} + {z^2}}$, respectively. $\tilde \pi$, $\tilde \tau$, and $\tilde P_n$ are the angular functions having the expressions as follows

(13)$$\begin{aligned} &\tilde P_n^m(\cos \theta ) = \frac{1}{{{2^n}n!}}{(\sin \theta )^m}\frac{{{d^{n + m}}}}{{{{(d\cos \theta )}^{n + m}}}}{\left( {{{\cos }^2}\theta - 1} \right)^n}\\ &\tilde \pi _n^m = \frac{{\tilde P_n^m}}{{\sin \theta }},{{\tilde \tau }_n} = \frac{{d\tilde P_n^m}}{{d\theta }} \end{aligned}$$

Substituting Eqs. (8)–(10) into Eq. (2), the magnetic fields are expressed as follows

(14)$${{\bf{H}}^{inc}}(r) ={-} {\rm{j}}\frac{{{k_{{m_2}}}}}{{\omega {\mu _0}}}\sum_{n = 1}^\infty {\sum_{m ={-} n}^n {{E_0}\left[ {{A_{mn}}{\bf{N}}_{mn}^{(1)}\left( {{k_{{m_2}}}r} \right) + {B_{mn}}{\bf{M}}_{mn}^{(1)}\left( {{k_{{m_2}}}r} \right)} \right]} }$$

(15)$${{\bf{H}}^{sca}}(r) = {\rm{j}}\frac{{{k_{{m_2}}}}}{{\omega {\mu _0}}}\sum_{n = 1}^\infty {\sum_{m ={-} n}^n {{E_0}} } \left[ {{a_{mn}}{\bf{N}}_{mn}^{(3)}({k_{{m_2}}}r) + {b_{mn}}{\bf{M}}_{mn}^{(3)}({k_{{m_2}}}r)} \right]$$

(16)$${\bf{H}^{{\mathop{\rm int}} }}(r) ={-} {E_0}\sum_{n = 1}^\infty {\sum_{m ={-} n}^n {\left\{ \begin{array}{l} {\rm{j}}\frac{{{\kappa _m}}}{{\omega {\mu _0}}}\left[ {{c_{mn}}{\bf{N}}_{mn}^{(1)}({\kappa _m}r) + {d_{mn}}{\bf{M}}_{mn}^{(1)}({\kappa _m}r)} \right]\\ + \left( {{\varepsilon _{\rm{2}}} - {\varepsilon _1}} \right)\Omega r\sin \theta \left\{ \begin{array}{l} \left[ {{c_{mn}}{M}_{mn}^{(1)}{{({\kappa _m}r)}_\theta } + {d_{mn}}{N}_{mn}^{(1)}{{({\kappa _m}r)}_\theta }} \right]{{\bf{i}}_r}\\ - {d_{mn}}{N}_{mn}^{(1)}{({\kappa _m}r)_r}{{\bf{i}}_\theta } \end{array} \right\} \end{array} \right\}} }$$

with the wave number of the electromagnetic field outside the sphere ${k_{{m_2}}} = {m_2}k$. ${\bf {i}}_r$ and ${\bf {i}}_\theta$ are the components of $\bf {i}$ in the $r$ and $\theta$ directions, respectively. The subscript $\theta$ and $r$ are the corresponding component expressions of VSWFs.

$A_{mn}$ and $B_{mn}$ are the incident expansion coefficients, which are derived according to the theory in [72] as follows

(17)$$\begin{aligned} &{A_{mn}} ={-} \frac{{4{{\rm{j}}^n}}}{{\sqrt {2n\left( {n + 1} \right)} }}\sqrt {\frac{{2n + 1}}{2} \cdot \frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \left[ \begin{array}{l} ({\rm{j}}m\cos \chi \cos {\alpha _{pol}} + m\sin \chi \sin {\alpha _{pol}})\tilde \pi _n^{|m|}(\beta )\\ + (\cos \chi \sin {\alpha _{pol}} + {\rm{j}}\sin \chi \cos {\alpha _{pol}})\tilde \tau _n^{|m|}(\beta ) \end{array} \right]{{\rm{e}}^{{\rm{ - j}}m\alpha }}\\ &{B_{mn}} ={-} \frac{{4{{\rm{j}}^n}}}{{\sqrt {2n\left( {n + 1} \right)} }}\sqrt {\frac{{2n + 1}}{2} \cdot \frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \left[ \begin{array}{l} ({\rm{j}}\cos \chi \cos {\alpha _{pol}} + \sin \chi \sin {\alpha _{pol}})\tilde \tau _n^{|m|}(\beta )\\ + (m\cos \chi \sin {\alpha _{pol}} + {\rm{j}}m\sin \chi \cos {\alpha _{pol}})\tilde \pi _n^{|m|}\left( \beta \right) \end{array} \right]{{\rm{e}}^{{\rm{ - j}}m\alpha }} \end{aligned}$$

Equation (17) defines an arbitrary optical polarized plane wave. Where the transmission direction is decided by $\alpha$ and $\beta$, the polarization direction is controlled by orientation angle $\alpha _{pol}$, and the polarization states are determined by polarized ellipticity angle $\chi$ whose expression is $\chi = \pm \arctan \left ( {{b \mathord {\left /{\vphantom {b a}} \right.} a}} \right )$, $a$ and $b$ are the semi-axes of the vibration ellipse in the electromagnetic field vibration plane, respectively. Coefficients $a_{mn}$, $b_{mn}$, $c_{mn}$ and $d_{mn}$ are scattering expansion coefficients.

Considering the continuity of electromagnetic field at the boundary of a rotating dielectric sphere ($r=a$) as follows

(18)$$\left\{ \begin{array}{l} E_\theta ^{{\mathop{\rm int}} } = E_\theta ^{inc} + E_\theta ^{sca},E_\varphi ^{{\mathop{\rm int}} } = E_\varphi ^{inc} + E_\varphi ^{sca}\\ H_\theta ^{{\mathop{\rm int}} } = H_\theta ^{inc} + H_\theta ^{sca},H_\varphi ^{{\mathop{\rm int}} } = H_\varphi ^{inc} + H_\varphi ^{sca} \end{array} \right.$$

Substitute Eqs. (8)–(10) and Eqs. (14)–(16) into Eq. (18) and simplify, the following relationship is obtained

(19)$$\begin{aligned} &A{a_{mn}} + B{b_{mn}} + C{c_{mn}} + D{d_{mn}} = Q\\ &E{a_{mn}} + F{b_{mn}} + G{c_{mn}} + H{d_{mn}} = R\\ &I{a_{mn}} + J{b_{mn}} + K{c_{mn}} + L{d_{mn}} = S\\ &M{a_{mn}} + N{b_{mn}} + O{c_{mn}} + P{d_{mn}} = T \end{aligned}$$

where

(20)$$\begin{array}{cc} {{\rm{A = j}}m\tilde \pi _n^{|m|}(\theta )\frac{{\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}}{{{k_{{m_2}}}r}}} & {B = \tilde \tau _n^{|m|}(\theta )\frac{{{{\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }}}{{{k_{{m_2}}}r}}}\\ {C = {\rm{j}}m\tilde \pi _n^{|m|}(\theta )\frac{{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}}{{{\kappa _m}r}}} & {D = \tilde \tau _n^{|m|}(\theta )\frac{{{{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}^\prime }}}{{{\kappa _m}r}}}\\ {E ={-} \tilde \tau _n^{|m|}(\theta )\frac{{\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}}{{{k_{{m_2}}}r}}} & {F = {\rm{j}}m\tilde \pi _n^{|m|}(\theta )\frac{{{{\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }}}{{{k_{{m_2}}}r}}}\\ {G ={-} \tilde \tau _n^{|m|}(\theta )\frac{{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}}{{{\kappa _m}r}}} & {H = {\rm{j}}m\tilde \pi _n^{|m|}(\theta )\frac{{{{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}^\prime }}}{{{\kappa _m}r}}}\\ {I = \tilde \tau _n^{|m|}(\theta )\frac{{{{\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }}}{r}} & {J = {\rm{j}}m\tilde \pi _n^{|m|}(\theta )\frac{{\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}}{r}}\\ {K = \tilde \tau _n^{|m|}(\theta )\frac{{{{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}^\prime }}}{r}} & {L = \left[ \begin{array}{l} {\rm{j}}\left( {m_1^2 - m_2^2} \right)\gamma \left( {kr} \right)n(n + 1) \sqrt {1 - {{\cos }^2}\theta }\\ \times \frac{{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}}{{{{\left( {{\kappa _m}r} \right)}^2}}}\tilde P_n^{|m|}(\cos \theta ) + {\rm{j}}m\tilde \pi _n^{|m|}(\theta )\frac{{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}}{r} \end{array} \right]}\\ {M ={-} {\rm{j}}m\tilde \pi _n^{|m|}(\theta ){{\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }}&{N = \tilde \tau _n^{|m|}(\theta )\left[ {{k_{{m_2}}}rh_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}\\ {O ={-} {\rm{j}}m\tilde \pi _n^{|m|}(\theta ){{\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]}^\prime }}&{P = \tilde \tau _n^{|m|}(\theta )\left[ {{\kappa _m}rj_n^{\left( 1 \right)}({\kappa _m}r)} \right]} \end{array}$$

(21)$$\begin{array}{c} {{\rm{Q = j}}m\tilde \pi _n^{|m|}(\theta )\frac{{\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}}{{{k_{{m_2}}}r}}{A_{mn}} + \tilde \tau _n^{|m|}(\theta )\frac{{{{\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }}}{{{k_{{m_2}}}r}}{B_{mn}}}\\ {R ={-} \tilde \tau _n^{|m|}(\theta )\frac{{\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}}{{{k_{{m_2}}}r}}{A_{mn}} + {\rm{j}}m\tilde \pi _n^{|m|}(\theta )\frac{{{{\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }}}{{{k_{{m_2}}}r}}{B_{mn}}}\\ {S = \tilde \tau _n^{|m|}(\theta )\frac{{{{\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }}}{r}{A_{mn}} + {\rm{j}}m\tilde \pi _n^{|m|}(\theta )\frac{{\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}}{r}{B_{mn}}}\\ {T ={-} {\rm{j}}m\tilde \pi _n^{|m|}(\theta ){{\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]}^\prime }{A_{mn}} + \tilde \tau _n^{|m|}(\theta )\left[ {{k_{{m_2}}}rj_n^{\left( 1 \right)}({k_{{m_2}}}r)} \right]{B_{mn}}} \end{array}$$

The equation for the scattering expansion coefficients (Eq. (19)) can be solved using Cramer’s rule. Therefore, the expressions of total electric and magnetic fields are defined as follows

(22)$${{\bf{E}}^{total}} = {{\bf{E}}^{inc}} + {{\bf{E}}^{sca}} + {{\bf{E}}^{{\mathop{\rm int}} }}$$

(23)$${{\bf{H}}^{total}} = {{\bf{H}}^{inc}} + {{\bf{H}}^{sca}} + {{\bf{H}}^{{\mathop{\rm int}} }}$$

the superscript $total$ expresses the vector sum of the incident, scattered, and internal electromagnetic fields.

Ulteriorly, the time-averaged Poynting vector is expressed as

(24)$$\left\langle {\bf{S}} \right\rangle = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{{\bf{E}}^{total}} \times {{\bf{H}}^{total}}^{^ * }} \right\}$$

where $\left \langle {\cdots } \right \rangle$ and $*$ are the time-average and conjugate symbols, respectively. And ${\mathop {\rm Re}\nolimits }$ is the real part of the symbol.

The components of the time-averaged Poynting vector in the $x$-, $y$-, and $z$-direction are given as

(25)$$\begin{aligned} &{{\bf{S}}_x} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{\bf{E}}_y^{total}{\bf{H}}_z^{tota{l^ * }} - {\bf{E}}_z^{total}{\bf{H}}_y^{tota{l^ * }}} \right\}\\ &{{\bf{S}}_y} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{\bf{E}}_z^{total}{\bf{H}}_x^{tota{l^ * }} - {\bf{E}}_x^{total}{\bf{H}}_z^{tota{l^ * }}} \right\}\\ &{{\bf{S}}_z} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left\{ {{\bf{E}}_x^{total}{\bf{H}}_y^{tota{l^ * }} - {\bf{E}}_y^{total}{\bf{H}}_x^{tota{l^ * }}} \right\} \end{aligned}$$

where the subscripts $x$, $y$, and $z$ represent the $x$-, $y$-, and $z$-direction components of corresponding vectors.

3. Numerical results

This section calculates the electromagnetic field intensities that a counterclockwise rotating dielectric sphere (It corresponds to the case $\gamma >0$) located in a medium is illuminated by a polarized plane wave with wavelength $\lambda = 0.532$ ${\rm {\mu m}}$ to study the optical Magnus effect. Mie size parameter $ka=36.0782$. The PH, resonance, and the impact of resonance on PH are elaborated in detail, individually. In simulation results, the refractive index of a dielectric sphere and medium are set as 1.5195 and 1.00. The boundary of the particle is remarked by a red line, and a black line as an auxiliary line corresponding to the $y = 0$ axis to study the deviation degree of PHs. Based on the above electromagnetic field expressions, the relative intensities $I_{\rm E}^{\text {total }}=\left |\mathbf {E}^{\text {total }} / E_{0}\right |^{2}$ and $I_{\rm H}^{\text {total }}=\left |\mathbf {H}^{\text {total }} / H_{0}\right |^{2}$ and their components in different directions are numerically calculated. The normalized electromagnetic field amplitudes are $E_{0}=1 \mathrm {~V} / \mathrm {m}$ and $H_{0}=1 \mathrm {~A} / \mathrm {m}$. Moreover, the energy flow distribution in the $xy$ plane ${S_{xy}} = \left | {\sqrt {{{\left | {{{\bf {S}}_x}} \right |}^2} + {{\left | {{{\bf {S}}_y}} \right |}^2}} } \right |$ is also calculated to discuss the unusual phenomena caused by rotation. Although the cases of linear, circular, and elliptically polarized plane waves incident on a rotating dielectric sphere are all considered in the derivation of the previous section, only the scattering of a sphere to a $z$- polarized plane wave is calculated as an example to avoid confusion for the reader. It is worth noting that the polarization state of an incident beam also has a significant effect on the scattering of rotating particles, which will be discussed separately in later studies.

3.1 PH generated by a rotating dielectric sphere

As a kind of curved PNJ, the effective length $L_{jet}$, deviation angle $\psi$, and tilt angle $\vartheta$ of the PH, are determined by start, inflection, and end points, shown in Fig. 1(a). The inflection point is the position where the maximum intensity $I_{max}$ of PHs is located [40]. The start and end points are the positions where the $I_{max}/e$, which are distributed to the far left and far right of PHs. $L_{jet}$ is the line connecting the start, inflection, and end points. $\psi$ is the included angle between the incident beam and the line from the start to inflection points, which is used to describe the degree of deviation from the direction of the incident beam. $\vartheta$ defined to elaborate the degree of curvature [39] is the included angle between two lines, which are determined by the starting and inflection points, and the inflection and end points, respectively.

First of all, the scattering intensity of a dielectric sphere spinning at various angular velocities to a $z$- polarized plane wave is calculated in Fig. 2, (a)-(c) correspond to the cases where $\gamma$ is set as $2 \times {10^{ - 2}}$, $4 \times {10^{ - 2}}$, and $6 \times {10^{ - 2}}$, severally. It means that $\Omega$ is $1.9641 \times {10^{12}}$, $3.9283 \times {10^{12}}$, and $5.8924 \times {10^{12}}$ rps (rps is short for revolutions per second), respectively. It has been seen that more field intensity is concentrated in the upper half of the particle. It is mainly due to the rotation of particles causing the symmetry of the system to be broken and further introducing additional electromagnetic fields inside the sphere. Although a large convergence of field intensity appears in the forward direction (the transmission direction of the incident plane wave) of particles which is similar to the PNJ, the focal point (position of the maximum intensity of the field) gradually moves up the half plane along the direction of rotation based on the continuity condition of a particle surface as $\gamma$ increases. To further illustrate the influence of $\gamma$ on PHs, the zoom of Fig. 2 is given in Fig. 3. It has been found that the electric field near the particle surface clusters in a half-plane that corresponds to the direction in which a particle rotates, both inside and outside the particle. It needs to be stressed that it is not just a simple translation of the field along the direction of rotation, but a complex superposition of the field belonging to the two states of stationary and rotation, respectively. It has been verified in Fig. 3(a) and (c) that side lobes corresponding to the direction of a particle rotation have greater intensity. However, it should be distributed symmetrically along the $y=0$ axis if a sphere is stationary. Furthermore, the deviation angle $\psi$ is 14.3207$^\circ$, 24.9470$^\circ$, and 36.3263$^\circ$ in Fig. 3(a)-(c), respectively. Accordingly, the $\vartheta$ also changes as $\gamma$ increases, it is 2.0388$^\circ$, 2.7527$^\circ$, and 1.8861$^\circ$. It means the additional field caused by rotation causes the PNJ to curve, forming PH. Thus it is not unusual that $L_{jet}$ changes as $\gamma$ increases due to the appearance of a hook, it is 1.1925, 1.1476, and 1.1278 ${{\rm {\mu m}}}$, separately.

Fig. 2. The scattering intensities of a rotating dielectric sphere to a $z$- polarized plane wave, and the non-reciprocal rotating dimensionless parameter $\gamma$ is set as $2 \times {10^{ - 2}}$, $4 \times {10^{ - 2}}$, and $6 \times {10^{ - 2}}$ from (a)-(c), respectively.

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Fig. 3. The zoom that a rotating dielectric sphere is illuminated by a $z$- polarized plane wave. The parameters are the same as those in Fig. 2.

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Therefore, it is significant to explore the influence of $\gamma$ on deflection angle $\psi$ and tilt angle $\vartheta$ as shown in Fig. 4, which is beneficial for designing a PH. In Fig. 4, the $\psi$ maintains an upward trend as $\gamma$ increases. It can be seen that this trend is generally linear although it shows a wavy upward trend at the occasional as $\gamma$ changes. Thus a conclusion is attained that the deflection direction of PHs is positively correlated with $\gamma$, which is of great significance to realize the three-dimensional manipulation of microscopic particles. Surprisingly, $\vartheta$ does not keep an increasing trend as $\gamma$ increases, as $\psi$ does. It does not show any particular regularity although it also changes as $\gamma$ increases. It means that an additional field caused by rotation changes the structure of PNJ and generates PHs. However, it does not mean that the faster the spin angular velocity of a particle, the more conducive to the formation of PH with greater curvature, which also depends on a series of parameters such as the radius and the relative refractive index of a particle, and the wavelength of the incident wave. Compared with $\vartheta$, $\psi$ plays a more important role in designing a PH. It has promising application potential in super-resolution imaging and optical manipulation. The other characteristics of PHs, including the intensity of the focal point (focal point is defined as the point of maximum intensity within the PH) and $L_{jet}$, have nothing to do with the $\gamma$ and are not given here.

Fig. 4. The influence of rotation on the deflection angle $\psi$ and tilt angle $\vartheta$. $\gamma$ is chosen in the range from 0 to 0.06. The other parameters are the same as those in Fig. 2.

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3.2 Resonance generated by a rotating dielectric sphere

It has been verified that the asymmetry of a system generated by rotation plays a role in the PHs. As is known to all, the characteristic of the WGM resonance is sensitive to the refractive index and radius changes [76]. However, once the particle radius is determined, it is difficult to change it again to achieve resonance conditions, which limits the development of whispering-gallery resonators (WGRs). Although we have accidentally observed weak resonance caused by rotation in a previous study [42], whether it can be used in the design of resonators remains a mystery. If it can be used to design a resonator, it means that any dielectric material and spherical particles of any radius can interact with incident light of a single frequency, thus generating resonance scattering, which will greatly reduce the design difficulty and production cost of the resonator. Hence Fig. 5 is calculated to achieve this goal and provide new theoretical guidance for the design of resonators.

Fig. 5. The resonance generated by a rotating sphere. $\gamma$ is set as $8.3358 \times {10^{ - 3}}$, $1.846813 \times {10^{ - 2}}$, and $5.8016 \times {10^{ - 2}}$ from (a)-(c). The other parameters are the same as those in Fig. 2.

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In Fig. 5(a)-(c) and (d)-(f) are the electric and magnetic scattering intensities of the plane wave with the same wavelength incidents on a dielectric sphere rotating at different angular speeds. In the subgraph of each row, from left to right, $\gamma$ is $8.3358 \times {10^{ - 3}}$, $1.846813 \times {10^{ - 2}}$, and $5.8016 \times {10^{ - 2}}$, respectively. They correspond to the case that $\Omega$ is $8.1863 \times {10^{11}}$, $1.8137 \times {10^{12}}$, and $5.6976 \times {10^{12}}$ rps. Comparing the subgraphs corresponding to the upper and lower rows can be found that both the electric field resonance and magnetic field resonance appear inside the same particle with a steady angular velocity of rotation. Furthermore, the magnetic resonance has a higher intensity than that of the electric resonance. Comparing Fig. 5(a)-(c), another thing is found different resonance modes can be attained by changing the rotation speed of a sphere. The maximum resonance intensity is distributed on the surface of a particle in Fig. 5(c), but it is distributed inside a sphere near the inner surface shown as Fig. 5(a)-(b). Different from the WGM resonance whose point arrays are distributed independently in the resonance ring, the intensity is more uniform across the resonance ring in Fig. 5. As is known to all, resonance occurs inside the particle when an incident beam frequency matches the natural frequency of the particle, which is determined by the size parameter and relative refractive index of the dielectric particle. Therefore, resonance only appears when a plane wave interacts with a dielectric particle once it is produced. However, the frequency matching conditions between particles and incident waves have been changed due to the introduction of rotation, and this further causes resonance within a spinning sphere. The light should continuously reflect inside the particle and form a standing wave under resonance conditions, which is also the reason for the distribution of point arrays in resonance rings of the WGM. However, the rotation of particles causes the standing wave position to move along the direction of particle rotation, resulting in the gradual converging of adjacent point arrays in the same resonance ring shown in Fig. 5(c) and (e). However, this effect is not evident in the other subgraphs of Fig. 5 because adjacent points have fully merged. Compared with the WGM resonator, the resonance with uniform intensity distribution may help simplify the design and optimization of resonance devices, making it easier to control their operating characteristics. Besides, it also improves the stability, uniformity, and applicability of a resonance system.

The intensity of the resonance scattering caused by rotation is already large enough, and can even achieve the intensity of super-resonance [57,77] (intensity gains of the order $10^5$-$10^7$ for both magnetic and electric fields) inside a particle, as shown in Fig. 5(a) and (d). However, different from the whispering gallery mode resonance mentioned above, the super-resonance is caused by the high-order Fano resonance. The impact of rotation on the super-resonance is also a new subject worth discussing because microscopic particles are not always stationary, motion is the norm. To reveal whether a moving particle can also be used to design a super-resonator, Fig. 6 is given. Similar to Fig. 5, the subgraphs in the first and second rows correspond to the electric and magnetic fields, respectively. From left to right, the rotating dimensionless parameter $\gamma$ is set as $0$, $1 \times {10^{ - 6}}$, and $2 \times {10^{ - 6}}$.

Fig. 6. The influence of rotation on the super-resonance. $\gamma$ is set as 0, $1 \times {10^{ - 6}}$, and $2 \times {10^{ - 6}}$ from (a)-(c). The other parameters are the same as those in Fig. 2.

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In Fig. 6, the results in (a) and (d) are the same as those in [78], which correspond to electric and magnetic super-resonance, respectively. Two hotspots appear inside a sphere in Fig. 6(d). Different from the magnetic super-resonance to the electric super-resonance, there are also two low-intensity hotspots inside a particle in addition to the two maximum-intensity hotspots that exist at the boundary of the particle in Fig. 6(a). Both the hotspots of electric and magnetic super-resonance are distributed along the $y=0$ axis. Comparing Fig. 6(a)-(c) and (d)-(f), a phenomenon is found that the super-resonance (including resonance) is sensitive to the angular velocity of rotation in addition to the size parameter and relative refractive index of a dielectric particle. The high-order Fano resonance has been destroyed as $\gamma$ increases. The hotspots gradually diverge from the $y=0$ axis along the rotation direction of a particle and move further away from it, as $\gamma$ increases. It is mainly because due to the introduction of rotation, the light path of the beam inside a particle changes, which further causes the position of the standing wave formed by the reflected light to shift. However, the farther away from the $y=0$ axis, the lower the electromagnetic field intensity although the location of the resonance inside the particle has been changed in Fig. 6. This is because the intensity of hotspots is much greater than that of an adjacent location to the Fano resonance. The difference is that the intensity of adjacent resonance arrays of the WGM resonance is similar, which further causes them to gradually merge as the particles rotate. This is the reason why the uniform resonance ring appears in Fig. 5. Therefore, there is a conclusion that the resonance is sensitive to the spin velocity of a particle. For example, maximal electric field intensity decreases from $9.70 \times {10^5}$ at $\gamma =0$ to $9.00 \times {10^3}$ and $2.22 \times {10^3}$ for $\gamma =1 \times {10^{-6}}$ and $2 \times {10^{-6}}$, respectively. On the other hand, maximal magnetic field intensity decreases from $1.62 \times {10^7}$ for $\gamma =0$ to $1.36 \times {10^5}$ and $2.93 \times {10^4}$ for $\gamma =1 \times {10^{-6}}$ and $2 \times {10^{-6}}$, respectively. This tendency clearly shows that magnetic field intensity is more sensitive to rotation speed than the electric field intensity. The rotation characteristics of the particle have broad application prospects in designing resonators and avoiding unnecessary resonance.

3.3 Impact of resonance on PH

At present, a conclusion is attained that the asymmetry of the rotating system has something to do with the generation of PHs. Moreover, it can also be used to generate or destroy resonance scattering. The asymmetry is determined by the spin angular velocity of a particle, it is possible that both the PH and resonance scattering appear simultaneously in the process of particle rotation. It is well-known that the characteristic of PNJ is impacted by resonance scattering. Whether the resonance also takes effect on the PH, although they are all caused by the rotation of particles. To explore this question, Fig. 7 is calculated, where both PHs and resonance scattering exist in the same model. In other words, we can call it "incomplete resonance". In the subgraph of each row, $\gamma$ is chosen as $1.3421 \times {10^{ - 2}}$, $4.45 \times {10^{ - 2}}$, and $4.9065 \times {10^{ - 2}}$ from left to right (In other words, $\Omega$ is $1.3180 \times {10^{12}}$, $4.3702 \times {10^{12}}$, and $4.8186 \times {10^{12}}$ rps, respectively). In Fig. 7(b)-(c) and (e)-(f), the resonance ring is formed by an array of adjacent points that are relatively independent of each other. In Fig. 7(a) and (d), a merging trend of the adjacent points appears. However, a commonality is that the side lobe in the direction of particle rotation has greater intensity under resonance conditions. It can be seen that a PH with larger curvature appears in front of a rotating sphere in Fig. 7, even if its intensity is smaller than that inside a particle. On the one hand, in resonance conditions, the intensity difference between the side lobe on either side of the $y=0$ axis intensifies, further causing the asymmetry of the field caused by rotation to become more pronounced. On the other hand, as $\gamma$ increases, the PH moves above the $y=0$ axis, which is more susceptible to the upper side lobe. Under the condition of "incomplete" resonance inside the particle, the interference effect of the main lobe and the upper side lobe is strengthened. Therefore, a conclusion has come up that the curvature of PHs is determined by the intensity of the side lobe along the direction of particle rotation, which is limited by the resonance intensity. In other words, the resonance inside particles caused by rotation plays an active role in producing PHs with greater curvature.

Fig. 7. The impact of resonance on the PH in the rotating system. $\gamma$ is set as $1.3421 \times {10^{ - 2}}$, $4.45 \times {10^{ - 2}}$, and $4.9065 \times {10^{ - 2}}$ from (a)-(c). The other parameters are the same as those in Fig. 2.

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Now, let us describe the "incomplete" resonance from the energy perspective. Fig. 8 is the ${\log_{10}}\left |{\sqrt {{{\left|{{{\mathbf {S}}_x}} \right|}^2}+{{\left|{{{\mathbf{S}}_y}}\right|}^2}}}\right |$ which calculates the distribution of energy when the "incomplete" resonance generates. The parameters are the same as those in Fig. 7(b). In Fig. 8, the visible vortices corresponding to the point arrays of resonance appear near the inner surface, which means more energy is concentrated here. The vortices, which are related to the formation of local areas with high values of local wave vectors [79], are surrounded by dense energy lines with circular distribution, which indicates that the energy of the resonance array tends to flow to the adjacent resonant array along the rotation direction of particles. It explains the formation of resonance in terms of energy. A phenomenon appears in Fig. 8 that the closer the edge of the particle, the denser the contours. However, the difference is that three long strip vortexes appear in front of the particle, which corresponds to the main and two side lobes of PH. More energy of the PH is concentrated above the $y=0$ axis since particles spin counterclockwise. Although the distribution of energy characterizes the PH and resonance ring under the "incomplete" resonance condition, the influence of "incomplete" resonance on the PH cannot be observed in Fig. 8.

Fig. 8. The ${\log_{10}}\left |{\sqrt {{{\left|{{{\mathbf{S}}_x}} \right|}^2}+{{\left|{{{\mathbf{S}}_y}}\right|}^2}}}\right |$ that the "incomplete" resonance appears. The parameters are the same as those in Fig. 7 and $\gamma =4.45 \times {10^{ - 2}}$.

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Therefore, the zoom of Fig. 7 is calculated to help analyze the influence of the "incomplete" resonance on PHs, shown as Fig. 9. The maximum intensity of the magnetic field is limited to 260 for observing the characteristics of PH to avoid the resonance intensity inside the particle being too large to cause the external PH shape to be unobtainable. In Fig. 9, the PH with greater focal point intensity and curvature emerges compared with Fig. 3. Although the deviation angle $\psi$ is little affected by resonance inside the spin particle, the tilt angle $\vartheta$ in Fig. 9 is $8.5911^\circ$, $32.6333^\circ$, and $11.1567^\circ$, which is much larger than that in Fig. 3. Homoplastically, the $\vartheta$ also is larger for the same reason in the magnetic field cases shown in Fig. 9(d)-(f). Therefore, a conclusion is attained that the high-intensity resonance generated inside the particle will further aggravate the bending properties of PHs under the resonance conditions caused by rotation. The influence of resonance on the focal point intensity, FWHM, and $L_{jet}$ has been extensively studied [65,68], thus it is not the end of this article.

Fig. 9. Zoom of the Fig. 7. The parameters are the same as those in Fig. 7.

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To further elaborate on the resonance and PH generated by rotation, the $S_{xy}$ and its zoom are discussed in Fig. 10(a)-(b), respectively. In Fig. 10(a), an interesting phenomenon is discovered that the energy distribution inside a particle is similar to the intensity distribution of the magnetic field shown in Fig. 7(e). Thus, the magnetic field determines the distribution of energy inside a particle. Besides, the flow direction of energy inside the sphere is clockwise, which is opposite to the rotation direction of a particle. It’s not surprising that more energy is concentrated in the region where resonance scattering occurs. However, an interesting thing is that the energies of adjacent point arrays tend to merge on the resonance ring. It also explains the reason for the uniform distribution of electromagnetic field intensity on the resonance ring inside a particle in terms of energy. A phenomenon is presented that the closer the edge of the particle, the higher the energy density. It means that more energy converges on the inner surface of particles due to the influence of resonance scattering. The huge energy distributed on the inner surface plays a critical role in the PH outside the particle, as shown in Fig. 10(b). It can be seen that the energy distribution of the resonance on the inner surface of the particle is non-uniform. More energy is distributed at the side lobe located in the upper half-plane and the junction of PH and the particle, which means that the higher energy density here. The energy inside a sphere flows clockwise, thus as shown by the arrows in Fig. 10(b), it flows from inside the particle to outside where the density is higher and converges at the front of a particle to generate a visible PH. In the junction of PH and the particle, the arrows start out pointing in the lower right direction, and as $x$ increases, they change direction and end up pointing in the upper right direction. This process provides theoretical support for the formation of PH caused by resonance. Therefore, a conclusion is proposed rotation can be utilized to introduce resonance scattering, which is beneficial for PH generation.

Fig. 10. The distribution and flow trend of energy when a plane wave exerts on a dielectric sphere rotating with $\gamma =4.45 \times {10^{ - 2}}$.

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4. Conclusion

Based on the optical Magnus effect, a conclusion is proposed in this paper that the spinning velocity causes the asymmetry of a system which is equivalent to introducing a transverse additional field due to particle deformation caused by the rotating. The rotation changes the characteristics of the PNJ and WGM resonance and further generates unusual optical phenomena, including the PH and the resonance with uniform intensity distribution on the ring. Besides, the bending characteristic of PH changed by rotation is first considered, it is also original and novel.

Under the influence of rotation, more energy is concentrated in the side lobe of the PNJ along the rotation direction of a particle, which further affects the main lobe. It provides a new method for generating PH. Although the deviation angle $\psi$ of PHs are linearly positively correlated to the non-reciprocal rotating dimensionless parameter $\gamma$, the effective length $L_{jet}$ and tilt angle $\vartheta$ is non-sensitive to the change of $\gamma$. Moreover, the asymmetry caused by rotation also is available for the creation and destruction of resonance scattering inside a sphere since the introduction of additional fields generated by rotation. The resonance scattering is sensitive to the $\gamma$, which has an extensive prospect in the spectrum analysis and design of the resonator. Besides, the influence of rotation on super-resonance is unique since a high-order Fano resonance with giant field enhancement exists only in a spherical particle rather than the cylindrical particle, which provides the theoretical basis for the optimization of super-resonator using rotation.

More interestingly, the internal resonance scattering makes sense to produce a PH with greater curvature outside the sphere. Due to the continuity of the field being determined by the direction of energy flow at the boundary of the rotating particle, the enhancement of the internal field by resonance scattering leads to the enhancement of the side lobe along the direction of rotation of the particle. It interacts with the main lobe of PNJ and is further beneficial for the production of PH, both its intensity and curvature are impacted. Therefore, the influence of resonance should also be considered when PH is designed.

The above conclusions are also applicable to other cases in which spin dielectric spheres scattering other polarized plane waves. Due to space limitations, we will not go into details here. In addition to the areas like resonance and particle manipulation mentioned and discussed emphatically in this paper, the theoretical framework of the scattering by spinning dielectric sphere to polarized plane wave also has extensive application prospects in the fields of planetary exploration and the in-depth exploration of plasma. No more than, our theory can be extended to the interaction between a shaped beam with a coated/multilayered sphere [80], which is useful for the development of optical tweezers.

Funding

Fundamental Research Funds for the Central Universities (YJSJ23017); National Natural Science Foundation of China (62001345, 62201411, 92052106).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (62001345, 62201411,92052106), and Fundamental Research Funds for the Central Universities (YJSJ23017). The research was partially supported by the Tomsk Polytechnic University Development Program.

Disclosures

The author declares 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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Unusual optical phenomena inside and near a rotating sphere: the photonic hook and resonance

Abstract

1. Introduction

2. Scattering by a rotating dielectric sphere

3. Numerical results

3.1 PH generated by a rotating dielectric sphere

3.2 Resonance generated by a rotating dielectric sphere

3.3 Impact of resonance on PH

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (25)

Optics Express