1. Introduction

As is known to all, only the longitudinal optical radiation force (OF) component exists when a symmetrical homogeneous isotropic dielectric particle static suspended in free space is illuminated by the axisymmetric incident beam. The transverse component of the OF disappears because of the symmetry of the incident beam and the particle. A few researchers have discussed this phenomenon, such as the interaction between a plane wave and a stationary cylindrical particle [1,2] or a spherical particle [1,3–5], and the interaction between a Bessel beam with a stationary symmetrical particle on the axis [6,7]. However, symmetry can be broken up when a structural beam interacts with particles [8–10]. Besides, replacing symmetrical particles with asymmetric particles also is a common method to break up symmetry [11,12].

The transverse component of the force will appear when the particle spins around its central axis, even if both the incident wave and the particle are symmetrical [13–15]. Because it is similar to the Magnus effect in hydrodynamics, the asymmetrical electromagnetic field generated by the rotating of the particle is considered the electromagnetic analog of the Magnus effect (optical/electromagnetic Magnus effect) [16,17]. The optical/electromagnetic Magnus effect has been widely studied and discussed [18–22]. A series of research results have been published, involving the fields of scattering [23], optical radiation force, and torque [24]. The research explored the electromagnetic Magnus effect produced by a rotating conductor cylinder [25] at a constant rotational angular velocity, and the scattering pattern and backscattering cross section were highlighted. Another research focused on the electromagnetic Magnus effect based on the interaction between a rotating dielectric cylinder [26]. Besides, the scattering of a rotating electron-plasma column to an electromagnetic wave was also analyzed [27]. After that, more attention was spent on the electromagnetic Magnus effect when a rotating sphere particle is illuminated by a plane wave [28]. And a study summarized the interaction between rotating particles and an electromagnetic field, which provided sufficient theoretical support for the analysis of the low rotational angular velocity (rotating dimensionless parameter $\beta <<1$) electromagnetic Magnus effect [29]. Although the Magnus effect due to the interaction between low rotational angular velocity particles and an electromagnetic field is more realistic, a particle with high rotational angular velocity interacting with an electromagnetic field is also worth investigating. The relativistic solution to the scattering problem by a rotating dielectric column [30] was given, which discussed the situation of particles rotating at a higher angular velocity. Not only that, the system analysis of the optical Magnus effect and detailed derivation in various situations were given [31]. What’s more, the optical Magnus effect with Berry phases and the Aharonov-Bohm effect was shown in previous research [21,32,33]. Recently, a study discussed the transverse spin and orbital angular momentum having the aid of the Magnus effect [34]. And the optical Magnus effect was expanded to the optical radiation force and torque by Mitri [24]. Besides, the optical Magnus effect in the photophoresis of a spinning absorptive dielectric circular cylinder was attained [23]. Similar to the electromagnetic Magnus effect, the Sagnac effect [35] also induces light deflection. It has been widely studied [36,37] and applied in daily life, such as fiber optic gyroscopes [38], Global Positioning Systems [39], and coupled-resonator [40].

Meanwhile, the photonic nanojet (PNJ) [41] has been another research hotspot because of its unique characteristic and various applications [42–44]. The PNJ generated, which enhances the forward of visible light by nanometer-scale dielectric particles by several orders of magnitude, it is a scattered beam of light with a high-intensity main lobe, a weak sub-diffracting waist, and a shallow divergence angle [45] (the propagation direction of incident light is defined as the forward direction in this manuscript). The simulation PNJ was initially generated by making use of the finite difference time domain (FDTD) after PNJ was found in a study [45]. And then, the Mie theory [46] was used to solve the scattered field to study the PNJ is also a helpful method. In another research [47], the Mie theory solution was decomposed into a Debye series to analyze the phenomenon in the PNJ research process. It has been proved that PNJ results from the joint action of particle shape size, relative refractive index, and incident wavelength. After that, another study used the multipole method to calculate the optical force generated by PNJ, and theoretically studied the influence of the radius and relative refractive index of the micro cylinder, and the polarization states of a plane wave on PNJ [48]. Besides, the PNJ is produced by particles of different shapes that also come into people’s sight, such as cuboids [49], hemispherical particles [50], microspheres with concentric rings [51], and pyramid [52–54]. More than this, a few structured beams were used as incident sources to generate PNJ in recent years, such as the Bessel beam [55], Gaussian beam [56], and the polychromatic beam [57].

Despite the plenty of research on rotating particles and PNJ, few people combine them for research. Similar to the PNJ, the curved photonic nanojet (CPNJ) has many application prospects in optical tweezers [58,59] and particle manipulation [60,61]. In addition, it can change the direction of nanojet, which has its advantage over PNJ. Therefore, this manuscript focuses on the optical Magnus effect to study the CPNJ produced due to the interaction between a dielectric circular cylinder rotating at a stable angular velocity and a plane wave for the above purposes. In the second section, the analytical expressions of electric field components under TE and TM polarization states are given using the partial-wave series expansion method [62,63] in cylindrical coordinates. The scattering coefficients are derived after considering the appropriate boundary conditions based on the instantaneous rest-frame theory [14], and the expression of scattered fields is obtained. In the third section, the numerical results of CPNJ are given. The influence of size parameter $ka$, relative refractive index $m$, and rotating dimensionless parameter $\beta$ are discussed in this process. The fourth section summarizes the work of the whole manuscript.

2. Theoretical background

An off-axis CPNJ is generated when a non-magnetic infinitely long dielectric circular cylinder rotating in a stable angular is illuminated by a plane wave propagating along $x$ direction, which is presented in Fig. 1. In this demonstration diagram, the circular cylinder of radius $a$ is located in the cylindrical coordinate system $O- r\theta z$ and is surrounded by the vacuum ambient medium. The angular velocity $\Omega _0 << c/a$ is based on the instantaneous rest-frame theory [25,64], where $c$ is the speed of light in free space.

 

Fig. 1. raphical representation for the curved photonic nanojet (CPNJ) generated by the interaction of a TM polarized incident wave with a dielectric cylinder spinning around the z-axis with an initial angular velocity $\Omega _0$. The rotating cylinder is located in the cylindrical system of coordinates $(r, \theta, z)$ whose center is $O$.

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The axial component of the TM polarization incident electric field can be presented as follows according to the partial-wave series expansion method in cylindrical coordinates based on the instantaneous rest-frame theory when the ${e^{ - i\omega t}}$ is chosen as the time-dependence factor with $\omega$ being the angular frequency.

After the interaction between the incident electric field and the stationary dielectric circular cylinder, the analytical form of the axial scattered electric field is attained according to the Helmholtz equation as follows

The internal axial electric field $E_z^{\operatorname {int} {\text {TM}}}$ in the cylinder must satisfy the following expression [14]

The expression of ${E_z^{\operatorname {int} {\text {TM}}}}$ given by [14,64] when a dielectric circular cylinder rotates with a low angular velocity.

After considering the boundary conditions at $r = a$, the scattering coefficients are obtained.

Similar to the derivation of electromagnetic fields under the TM polarization state, those expressions under the TE polarization state are easy to be derived, and are given in Appendix A. The scattering coefficients in this situation are solved as

Therefore, total electric field intensities $I^{TM}_{total}$ and $I^{TE}_{total}$ are written as

3. Numerical results and discussions

The numerical results of the CPNJ generated by the interaction between a plane wave (TM and TE) and a spinning dielectric circular cylinder with low angular velocity according to Eq. (8) and Eq. (9) are analyzed and discussed in this section, separately. And the maximum integer limit of partial-wave series is ${N_{\max }} = \operatorname {int} \left [ {ka + 4.05{{(ka)}^{1/3}} + 2} \right ]$, where the int expresses integer.

The total electric field intensities $I_{total}^{{\text {TM}}}$ and $I_{total}^{{\text {TE}}}$ are calculated. Where the electric field amplitude $E_0 = 1 V$, the relative refractive index of the particle is $m = 1.33$, and the wavelength of the incident wavelength is $\lambda = 632.8 nm$. In addition, to meet the instantaneous rest-frame theory $\beta << 1$, the $\beta$ is chosen in the range 0 - 0.06. Nonetheless, it must be clear that applications of this theory are far away because it is difficult to achieve such a particle speed under current conditions after a reasonable cylinder radius is chosen. And this manuscript highlights the influence of $\beta$, which determines the curved degree of CPNJ. And the curved degree is defined as the distance between the forefront of PNJ and $y = 0$. In addition, the rotation direction of CPNJ is symmetric about the $z$-axis when the symbol of $\beta$ is opposite because of the non-reciprocal of this system. This manuscript discusses the CPNJ as $\beta > 0$ as an example.

Let’s focus on the PNJ when the rotating dimensionless parameter $\beta$ is zero as a reference and comparison before discussing the CPNJ. The $I_{total}^{{\text {TM}}}$ and $I_{total}^{{\text {TE}}}$ are calculated and depicted in Fig. 2. The size parameter and relative refractive index of the static dielectric circular cylinder are $ka = 20$ and $m = 1.33$, respectively. The incident beam can be the TM and TE polarized plane waves that are presented in Fig. 2(a) and (b). And the particle is marked with a white line in the figure.

 

Fig. 2. The total electric field intensities between a static dielectric circular cylinder with size parameter $ka = 20$ and the TM and TE plane waves. And the relative refractive index $m = 1.33$, the rotating dimensionless parameter $\beta$ is zero, respectively.

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There is a commonality between panels (a) and (b) of Fig. 2. It can be seen that the distribution of PNJ in the $kx - ky$ plane is symmetrical with the axis of symmetry $y = 0$, which has been drawn in the figure. However, some differences are shown between Fig. 2(a) and (b). On the one hand, the distribution of the scattered field is different, and a more substantial backscattering is obvious in Fig. 2(a) compared with (b). And there is a series of side lobes that appear in the forward scattering of (a). Not only that, the internal field is more apparent in (a). However, the full width at half maxima (FWHM) of the PNJ in (b) is broader, and the focal point (the position corresponding to the electric field maximum intensity) is farther away from the particle. On the other hand, the intensity of the scattered field generated by the interaction between the dielectric circular cylinder and the TM and TE polarization waves also is dissimilar. The maximum intensity of the PNJ in (a) is more excellent than that of (b).

After the situation that $\beta = 0$ is discussed, another situation that the CPNJ is generated as shown in Fig. 3(a) - (d) when $\beta$ is non-zero is analyzed. The size parameter and the relative refractive index of the circular cylinder are $ka = 20$ and $m = 1.33$. What’s more, the rotating dimensionless parameter $\beta$ is highlighted below as the heaviest influencing factor in determining the rotation of a circular cylinder. To satisfy the instantaneous rest-frame theory ($\beta << 1$), the $\beta$ is chosen as 0.03 and 0.06, severally. It means that $\Omega _{0}$ is $4.47 \times {10^{12}}{{rad} \mathord {\left / {\vphantom {{rad} s}} \right. } s}$ and $8.93 \times {10^{12}}{{rad} \mathord {\left / {\vphantom {{rad} s}} \right. } s}$, respectively. And the corresponding particle surface velocity is $9 \times {10^6}{m \mathord {\left / {\vphantom {m s}} \right. } s}$ and $1.8 \times {10^7}{m \mathord {\left / {\vphantom {m s}} \right. } s}$. It can see that the symmetry of the PNJ is broken up, and the focal point of PNJ doesn’t appear in the $y = 0$ but rather in the plane that $y < 0$, and a CPNJ appears although this phenomenon is not particularly obvious in Fig. 3(a) and (c). We define this phenomenon as "photonic nanojet curved". Of course, the internal field also is asymmetry, which can be explained by the continuity conditions. However, the curved degree of CPNJ gradually strengthens as the increase of $\beta$ comparing Fig. 3(a) -(b) and (c) - (d). The CPNJ almost completely deviates from $y = 0$ in Fig. 3(b) and (d). Nor is this all, a scattering enhancement occurs near the edge of the spinning dielectric circular cylinder illuminated by the TE plane wave. Compared with Fig. 2(d), in Fig. 3(d), the convergence of fields is generated by the asymmetry of field distribution because of the rotation of the dielectric circular cylinder, and the convergence center is located at (16.8, 5.4) in the $kx - ky$ plane. This exciting finding can be applied to particle manipulation in the plane of $xy$ compared with the PNJ only manipulating particles in the $x$ direction. Moreover, the FWHM and effective intensity (the ratio of total field intensity to that of incident plane wave field ${\left ( {{{{E_{total}}} \mathord {\left / {\vphantom {{{E_{total}}} {{E_0}}}} \right. } {{E_0}}}} \right )^2}$) of the CPNJ are hardly affected by $\beta$.

 

Fig. 3. The same as in Fig. 2, but $\beta = 0.03 , 0.06$ expressing the dielectric circular cylinder is spinning in a stable angular velocity.

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In addition to the rotating dimensionless parameter, the size parameter $ka$ also plays a significant role in the "photonic nanojet curved", especially to the curved degree of CPNJ. To explore this characteristic, the situation that a spinning dielectric circular cylinder with size parameter $ka = 250$ and relative refractive index $m = 1.33$ illuminated by TM and TE plane waves is calculated, as shown in Fig. 4. The choice of $\beta$ is the same as Fig. 3, but the corresponding $\Omega _{0}$ becomes $3.57 \times {10^{11}}{{rad} \mathord {\left / {\vphantom {{rad} s}} \right. } s}$ and $7.14 \times {10^{11}}{{rad} \mathord {\left / {\vphantom {{rad} s}} \right. } s}$, respectively. It can see that the jet distance of the CPNJ becomes longer comparing Fig. 4 with Fig. 3 as $ka$ increases. In addition, the CPNJ has a more vigorous effective intensity when a big particle is illuminated by the TM and TE plane waves. Indeed, the most important thing is that the curved degree of CPNJ increases as $ka$ increases by comparing Fig. 4 and Fig. 3. The focal point of the CPNJ has wholly disengaged from $y = 0$ in Fig. 4(a) and (c). When the rotating dimensionless parameter $\beta$ increases to 0.06, the forefront of the CPNJ ($ky = 70$) has a certain distance to the axis $y = 0$. It has been presented in Fig. 4(b) and (d). Although both the effective intensity and distribution of the CPNJ in Fig. 4(a) - (c) are similar, there is a novel discovery in Fig. 4(d) that the distribution of the internal field is extraordinary. A series of more vigorous electric fields are distributed in the internal of the particle in the form of a circular array with radius $\left | {{\text {kr}}} \right |$ from 214.5 to 250. The reason why the circle array appears is the resonance scattering, which induces the maximum scattering intensity generated in the resonance circle. Therefore, there is a meaningful conclusion that the spinning of the dielectric circular cylinder can cause and change resonance scattering, which can be used to design ultra-sensitive sensors and resonant cavities. Besides, a CPNJ with a longer effective length (the effective length is defined as the position between two points where the intensity of the focal point divides ${{e^2}}$) is produced with the increasing of $ka$, although it is short relative to the particle radius by comparing Fig. 3 and Fig. 4. It has promising application prospects in particle manipulation and particle trapping by adjusting the radius of the particle to attain an appropriate CPNJ.

 

Fig. 4. The same as in Fig. 3, but $ka = 250$.

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After considering the influence of $ka$ on PNJ, let’s pay attention to another vital factor, the relative refractive index $m$, which also has something to do with the CPNJ. Fig. 5 gives the influence of $m$. The settings of all parameters are consistent with those in Fig. 4 except for the $m$, which is set as 1.57. As is known to all, the relative refractive index $m$ of the particle can change the convergence effect of the fields, which will further affect the formation of CPNJ. The bigger the $m$, the more vigorous the forward convergence of particles. Therefore, a CPNJ with more vigorous intensity generates in Fig. 5(a) - (d) compared with those in Fig. 4. However, greater forward convergence will cause a CPNJ with a shorter effective length. How to choose between a more vigorous effective intensity and a farther effective length depends on what purpose the readers operate particles for. As our most concerned point, how the $m$ takes affect the curved degree of CPNJ deserves to be explored. It is discussed by comparing the corresponding graph Fig. 4 and Fig. 5. More side lobes appear in the negative space $ky < 0$ in Fig. 5(a) and (c). Not only that, the included angle between the line of focus and $y = 0$ becomes more extensive and the CPNJ with all its side lobes is distributed in the negative space in Fig. 5(b) and (d). Correspondingly, the weight of the negative space fields in the particle internal also is greater. Based on this characteristic, we can attain a CPNJ with a more vigorous curved degree using a medium with a more significant relative refractive index $m$, even though the instantaneous rest-frame theory must be considered that $\beta << 1$. This is helpful in the application fields of biomedical and photonics. Besides, the resonance scattering generated in Fig. 4(d) has disappeared in Fig. 5(d).

 

Fig. 5. The same as in Fig. 4, but $m = 1.57$.

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4. Conclusion

In conclusion, the analytical expressions of the electric field are derived from that a plane wave interacts with a spinning dielectric circular cylinder with a stable angular velocity $\Omega _0$. The instantaneous rest-frame theory and the partial-wave series expansion method in cylindrical coordinates are used in this process. The CPNJ generated by the spinning of the dielectric circular cylinder is discussed, and the influence of the rotating dimensionless parameter $\beta$ on PNJ is concentrated. In addition, the impact of size parameter $ka$ and the relative refractive index $m$ of the particle to the CPNJ is also discussed and analyzed.

As a crucial parameter that decides the spinning angular velocity, $\beta$ not only breaks up the symmetry of the electromagnetic field distribution but also determines the curved degree of CPNJ. The curved degree of CPNJ strengthens as $\beta$ increases, but it must be noted that $\beta << 1$ satisfies the instantaneous rest-frame theory. Additionally, $\beta$ can be chosen to produce and change the resonance scattering. Besides, because of the non-reciprocal of this system, it has promising applications in the non-reciprocal system in the future. Not only that, $ka$ and $m$ also play significant roles in the generation of CPNJ. The effective intensity and the effective length of the CPNJ increase as $ka$ increases. Moreover, the curved degree also gets strengthened as $ka$ increases. $m$ intensifies the convergence and maintains the effective intensity of the CPNJ. Moreover, it also enhances the curved degree of CPNJ. Although it is still a long time before the implementation of this research result, all the findings discussed and analyzed in this manuscript will play a significant role in particle manipulation, optical tweezers, and the design of resonators.