Mechanical effect of photonic spin-orbit interaction for a metallic nanohelix

Jun Chen; Shubo Wang; Xiao Li; Jack Ng

doi:10.1364/OE.26.027694

1. Introduction

Light-induced optical force and torque can be used to manipulate small particles. Such techniques have already found fruitful applications across different fields in fundamental and applied science [1–15]. While optical force is a consequence of linear momentum transfers [16–18], optical torque is due to the angular momentum transfer between the light and the illuminated object [19–26]. Light can carry two kinds of angular momentum: spin angular momentum (SAM) and orbital angular momentum (OAM). The SAM is defined by the polarization states of light, and each photon may carry a SAM of $+ ℏ$ or $- ℏ$ , corresponding to the left-handed (LH) and right-handed (RH) circular polarizations, respectively. The OAM is defined by the light’s spatial phase structure, i.e. the propagation of light. Within a homogeneous and isotropic environment, SAM and OAM are separately conserved quantities [27]. However, the presence of a particle breaks the symmetry and the corresponding conservation laws, which allows SAM and OAM to be transformed into each other that called spin-orbit interaction (SOI) of light [28–32]. Here, using full-wave simulations and analytic techniques, we show that the SOI of light can induce positive or negative optical torque on a simple metallic nanohelix (by “negative” we mean the direction of the torque is opposite to that of the incident SAM). We note that negative optical torque was predicted previously for spheroidal particles [33], structure possessing discrete rotational symmetries [34], and observed experimentally [32,35]. Here we show that the screw-axis symmetry can also allow such phenomenon, and that it is over a broad frequency range. The nanohelix can convert an incident circularly polarized plane wave into fields with different OAM. We show that the conversion is selective and depends solely on the intrinsic eigenmodes of the helix. Moreover, we found that spin-orbital angular momentum conversion is a necessary condition for inducing recoil optical torque, but not for light extinction.

We consider a RH metallic nanohelix (we remark that it is the right circularly polarized light that has the same helicity as the RH metallic nanohelix) consists of four pitches with inner radius r = 50 nm and outer radius R = 100 nm, as shown in Fig. 1(a). The helix is made of gold whose relative permittivity is described by the Drude model $ε_{Au} = 1 - ω_{p}^{2} / (ω^{2} + i ω ω_{t})$ , where $ω_{p} = 1.37 \times 10^{16} rad/s$ and $ω_{t} = 4.084 \times 10^{13} rad/s$ [36]. The structural chirality of the helix gives rise to many interesting phenomena such as lateral and pulling optical force [37,38], polarization conversion [39], chiro-metamaterials [40,41], etc. Here we are interested in the optical torque induced on the nanohelix by a circularly-polarized and z-propagating plane wave with an intensity of $1 {mW/μm}^{2}$ . We note that heating due to light absorption can be a serious problem for metallic particle, which could even destroy the particle at sufficiently high intensity. In the experiment in [24], a resonating plasmonic rotor (~200 nm in diameter) with a size comparable with our nanohelix is exposed to an intensity of ~ $1 {mW/μm}^{2}$ without observable damage. We noted that the nano plasmonic rotor is in water and deposited on a glass micro-substrate, but our nanohelix is not. These factors may weaken the temperature rise of the plasmonic rotor itself. We remark that although our calculations are for nanohelix in air, the optical torque and the contribution from various modes of the nanohelix in water should be similar due to the high dielectric constant of gold (such that the dielectric constants for air and water are both considered small). For these reasons, we would expect our nanohelix to survive a similar intensity, justifying the use of $1 {mW/μm}^{2}$ in our calculations. Moreover, our calculated torque is directly proportional to light intensity, and similar physical effects predicted here can be observed at lower intensity.

Fig. 1 (a) A schematic of the nanohelix is drawn to scale. The inner and outer radii are r = 50 nm and R = 100 nm, respectively. The pitch length is p = 300 nm. The incident plane wave propagates along z direction with intensity of 1 $mW / {μm}^{2}$ . Optical torque exerted on the RH helix made of PEC/gold under different polarizations versus frequencies and also their corresponding wavelength are shown in (b).

Download Full Size | PDF

The paper is organized as follows. In Sec. 2, we first introduce the numerical and analytical methods: the partial wave channel analysis and the eigenmode theory. We then present our numerical results and our discussions on the optical torque acting on the helix and light extinction in Sec. 3. A summary is given in Sec. 4.

2. Theory

2.1 Calculation of optical torque

To evaluate the optical torque acting on the nanohelix by the incident plane wave, we first calculate the electromagnetic (EM) fields using a full-wave finite-element method package from COMSOL [42]. Then we integrate the Maxwell stress tensor via a surface integral, $\overset{\leftrightarrow}{T} = ε_{0} E E + μ_{0} H H - 1 / 2 (ε_{0} {| E |}^{2} + μ_{0} {| H |}^{2})$ as: $〈 Γ 〉 = \oint n \cdot (r \times 〈 \overset{\leftrightarrow}{T} 〉) d S$ , which gives the optical torque acting on the nanohelix. Here the integral is performed over a closed surface that encloses the nanohelix and n denotes the unit outward normal vector on the surface.

2.2 Partial wave channel analysis of optical torque

Our partial wave treatment is similar to that of [34]. In summary, by expanding the incident field and scattered field in terms of vector spherical wave functions (VSWFs), the torque can be expressed as [34]

Γ_{z,tot} = \frac{2 π ε_{0} {| E_{0} |}^{2}}{k^{3}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} m Re [p_{m l} a_{m l}^{*} + q_{m l} b_{m l}^{*} - ({| a_{m l} |}^{2} + {| b_{m l} |}^{2})],

where

p_{m l}

and

q_{m l}

(

a_{m l}

and

b_{m l}

) are partial wave expansion coefficients for incident (scattered) waves [34]. Here “*” denotes the complex conjugate. It can be shown that the scattered power can be expressed as

W_{sca}^{(m)} = A_{0} \sum_{l} ({| a_{m l} |}^{2} + {| b_{m l} |}^{2})

while the extinction power can be expressed as

W_{ext}^{(m)} = A_{0} \sum_{l} Re (p_{m l} a_{m l}^{*} + q_{m l} b_{m l}^{*}),

where

A_{0} = 2 π ε_{0} c {| E_{0} |}^{2} / k^{2}

. Thus we have

Γ_{z, int} = \frac{2 π ε_{0} {| E_{0} |}^{2}}{k^{3}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} m Re [p_{m l} a_{m l}^{*} + q_{m l} b_{m l}^{*}] = \sum_{m_{i}} \frac{m_{i} ℏ}{ℏ ω} W_{ext}^{(m_{i})},

Γ_{z,sca} = - \frac{2 π ε_{0} {| E_{0} |}^{2}}{k^{3}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} m ({| a_{m l} |}^{2} + {| b_{m l} |}^{2}) = - \sum_{m} \frac{m ℏ}{ℏ ω} W_{sca}^{(m)},

and

Γ_{z,tot} = Γ_{z,int} + Γ_{z,sca} = \sum_{m_{i}} \frac{m_{i} ℏ}{ℏ ω} W_{ext}^{(m_{i})} - \sum_{m} \frac{m ℏ}{ℏ ω} W_{sca}^{(m)}

Here

m_{i}

and

m

denote the azimuthal number for the incident and scattered waves, respectively.

Γ_{z,int}

and

Γ_{z,sca}

denote the interception torque and recoil torque, respectively.

W_{ext}^{(m_{i})} / ℏ ω

denotes the number of photons being intercepted. Clearly the interception torque

Γ_{z, int}

is a consequence of extracting angular momentum from the incident wave. Moreover,

Γ_{z, int}

has the same sign with the SAM of the incident field because their signs are both determined by

m_{i}

. The recoil torque for each channel is proportional to the angular momentum scattered out by the helix through that channel. Equation (4) indicates that the extraction of total angular momentum from the incident wave contributes to a positive torque when the SAM of the incident wave is positive, while positive scattering angular momentum contributes to a negative torque. The total optical torque is a net result of the two.

2.3 Secular equation

For an intuitive understanding of the SOI, we resort to the analytical solution of the helix eigenmodes. An infinitely long helix has a screw-axis symmetry denoted by $U (ρ, ϕ, z) = U (ρ, ϕ \pm 2 π Δ z / p, z + Δ z)$ , where U is a physical entity, $(ρ, ϕ, z)$ denotes cylindrical coordinate system, and the + (-) sign corresponds to the RH (LH) helix. The symmetry arises from the fact that a translation of $Δ z$ combined with a rotation of $2 π Δ z / p$ recovers the original setting of the helix. With this symmetry, the eigenmodes of the helix can be expanded by the basis functions $F_{l} (ρ, ϕ, z) = A_{l} (ρ) Exp [i l ϕ] Exp [i (k_{z} + 2 l π / p) z]$ defined in cylindrical coordinates $(ρ, ϕ, z)$ . Here $A_{l} (ρ)$ is a function of the modified Bessel functions I_l and K_l [43], $l$ is an integer number, and $k_{z}^{}$ is the Bloch wave vector along z direction [43,44]. The basis functions indicate that the helix eigenmodes can carry OAM with angular number l. We apply Sensiper’s tape helix theory to determine the eigenmodes [43]. The corresponding secular equation is expressed as

\sum_{l} [(k_{z}^{2} R^{2} - k^{2} R^{2} + \frac{l^{2} k^{2}}{τ_{l}^{2}} \cot^{2} ψ) K_{l} (τ_{l} R) I_{l} (τ_{l} R) + k^{2} R^{2} \cot^{2} ψ {K^{'}}_{l} (τ_{l} R) {I^{'}}_{l} (τ_{l} R)] J_{l} = 0,

where

τ_{l} = {[{(k_{z} + 2 l π / p)}^{2} - k^{2}]}^{1 / 2}

and

J_{l} = \sin (l π r) / (l π r)

[43]. The eigen EM fields and eigen frequencies can be determined by solving Eq. (5). We then evaluate the z-component SAM density as

{〈 S 〉}_{z} = σ ℏ = \frac{ℏ ω}{W} \int_{Ω} {〈 S_{e} 〉}_{z} + {〈 S_{m} 〉}_{z} d A = \frac{ℏ ω}{W} Re {[\int_{Ω} \frac{ε_{0}}{4 ω i} E^{*} \times E + \frac{μ_{0}}{4 ω i} H^{*} \times H d A]}_{z},

where

〈 S_{e} 〉

and

〈 S_{m} 〉

denote the electric and magnetic parts of the SAM density, respectively [37].

Ω = π R^{2} \times p

is the volume occupied by one pitch of the helix and W is the total field energy inside a unit cell.

3. Results and discussions

3.1. Partial wave channel analysis

The numerically calculated optical torque exerted on a RH helix by the incident circularly polarized plane wave is shown in Fig. 1(b), for both a gold and a perfect electric conductor (PEC) helix. For the gold helix case, we notice that if the plane wave is RH polarized (solid blue line), carrying negative SAM, the induced torque is always negative in the considered frequency range. However, if the plane wave is LH polarized (solid red line), carrying positive SAM, the torque can be either positive or negative with a sign change near 430 THz. If we neglect the imaginary part of $ε_{Au}$ so that the material is lossless, similar results are obtained, as illustrated by the dashed red line, which means that the torque is almost entirely due to the scattering of the EM wave. As the angular momentum transferred to the helix through material absorption has the same sign with that of the incident wave, the torque for the lossy gold case is globally shifted upward compared to the lossless case. We also see that the PEC helix has similar results (solid green and magenta lines) except for a blue shift of the spectrum. If we change the handedness of the helix and the polarization simultaneously, the optical torque will also change sign, as a consequence of the mirror symmetry. Figure 2 shows the optical torque for the gold helices with different pitches. The optical torque spectrum remains more or less unchanged as the pitch number is increased from N = 2 to N = 10. Consequently, it suffices for us to focus on the N = 4 case in the following discussions.

Fig. 2 Optical torque exerted on the lossless RH gold helices with different pitch number N under the incidence of a LH polarized plane wave.

Download Full Size | PDF

By using partial wave expansion introduced in Sec. 2, the z direction optical torque can be expressed as: $Γ_{z,tot} = Γ_{z,sca} + Γ_{z,int}$ , where $Γ_{z,sca}$ and $Γ_{z,int}$ are numerically calculated and plotted in Fig. 3. Figure 3(a) shows the results for the RH gold helix excited by the LH polarized plane wave, which carries positive SAM. We note that the recoil torque and the interception torque have opposite contributions to the total torque. The interception torque has the same sign as the spin of the incident wave, while the recoil torque induced by an angular momentum channel m always rotate the helix in a direction opposite to the direction of the angular momentum. The summed total optical torque undergoes a change of the sign at the resonance frequency of 430 THz, which corresponds to the resonance of the recoil and interception torques. Figure 3(b) shows the results for the excitation under the RH polarized plane wave, which carries negative SAM. We notice that the total torque is negative in the whole spectrum. The recoil torque becomes positive which is opposite to the spin of the incident wave, while the interception torque is negative. Note that in this case, the helix resonates at a different frequency ( $\sim 220 THz$ ). For comparison, we also show the results for the PEC case in Figs. 3(c) and 3(d). We see that the results are roughly similar to that of the gold helix except for some frequency shift of the resonance peaks. This again confirms that it is the scattering but not absorption that plays the dominate role in inducing the optical torque.

Fig. 3 The recoil torque, the interception torque, and the total optical torque are shown, respectively, for a RH gold/PEC helix under LH polarization [(a), (c)] and RH polarization [(b), (d)].

Download Full Size | PDF

In order to understand above behaviors of the interception and recoil torques, we resort to partial wave channel analysis based on multipole expansions with VSWFs [45–50]. The torque contributed by each angular momentum channel is calculated according to Eqs. (2)-(4) and the results are shown in Fig. 4 for the gold helix case. We have truncated to $| m | \leq 2$ as higher order terms are negligibly small for a nanohelix. We see that for the LH polarized plane wave, $Γ_{z, int}$ is attributed to channel m = 1 [Fig. 4(a)] because the incident wave carrying SAM with quantum number $σ = + 1$ and OAM with quantum number $l = 0$ , hence the total angular momentum can only be $m = l + σ = 1$ . Figure 4(b) shows that the recoil part $Γ_{z, sca}$ has two major channel contributions: m = 1, 2. We note that the m = 1 channel of $Γ_{z, int}$ and $Γ_{z, sca}$ almost cancel each other, leaving the m = 2 channel to be the major contribution of the negative torque. We will show later that m = 2 channel is attributed to the eigenmode of the helix. The total optical torque shown in Fig. 4(c) is just a sum of the results in (a) and (b) based on Eq. (4). For the RH polarization case, we obtain dramatically different results. Figure 4(d) shows that the interception torque now comes from channel m = −1 as the incident SAM is flipped. For the recoil torque, the main contribution comes from channels $m = \pm 1$ and they induce torque of different signs according to Eq. (3), as shown in Fig. 4(e). The results in Fig. 4(f) show that $m = \pm 1$ channels contribute to a negative torque which has the same sign as the incident SAM. The major difference of LH and RH excitations is that the optical torque changes sign due to resonance in the former case while it is negative for the whole considered spectrum in the latter case. This difference comes from different angular momentum channels m excited under different polarizations.

Fig. 4 Optical torque exerted on a RH gold helix contributed by individual partial wave channels under the LH polarization [(a)-(c)] and the RH polarization [(d)-(f)].

Download Full Size | PDF

Figure 5 depicted the contributions from individual partial wave channels to light extinction. Figures 5(a) and 5(d) show the total extinction power for the LH and RH polarization, respectively. The contribution to the scattered power from each partial wave channel are shown in Figs. 5(b) and 5(e), respectively. Here, we note that as discussed before absorption does not change the physics qualitatively, and hence will be omitted. Figures 5(c) and 5(f) show the ratio of energy scattered into different channels, i.e. the energy scattered out by a specific azimuthal channel divided by the extinction power at a particular frequency. Clearly, the $Δ m = 0$ channels, namely m = 1 and m = −1 for the LH and RH polarization, respectively, are dominant both in extinction and scattered power. We note that in these channels the angular momentum initially absorbed through interception torque is re-emitted at the same azimuthal channel, so the entire process causes no angular momentum transfers, as there is no spin to orbital angular momentum conversion. It is clear that the extinction can be quite independent of the torque.

Fig. 5 Partial wave analysis of light extinction and scattered power. (a) and (d): the extinction power (the total energy intercepted) of a RH gold helix under LH and RH polarization. (b) and (e): the energy scattered into different partial wave channels. (c) and (f): the percentage of intercepted energy scattered into each different channels.

Download Full Size | PDF

3.2. Spin-orbit interaction

We now resort to the mechanism of spin-orbit interaction of light to understand the origin of different m channels excited by the incident wave of different polarizations. Light spin-orbit interaction is frequently encountered in optical scattering systems [51,52]. For waves scattered by a particle of arbitrary geometric shape, the scattered field generally has a non-uniform distribution of phase and amplitude in space. Therefore, mutual conversion between the SAM and OAM usually takes place in a scattering process. When circularly polarized plane waves impinges the nanohelix, the SAM of incident wave can be converted to the SAM and OAM of scattered wave. The conversion is achieved through the excitation of the eigenmodes of the helix.

For simplicity, we assume the helix to be made of PEC, and the physics is also applicable to the gold helix. The eigenmodes and band structure of the helix are determined by the secular equation of Eq. (5). The band structure is plotted in Fig. 6 as the solid and dashed lines. For comparison, we also presented band structure computed from the full-wave simulations by using COMSOL, where we calculate for a two-dimensional square lattice of the helix with a large period of $d = 4 μm$ . In this case, the coupling between neighboring helices is negligible and the band structure resembles that of a single helix. A good agreement between the analytical results and the numerical results is achieved. The gray region in Fig. 6 defines the light cone. Accordingly, the dashed lines within the cone correspond to radiating modes, while the solid lines outside the cone correspond to non-radiating guided modes. For a finite helix excited by an incident wave, both the radiating modes and the guided modes can be excited. However, the radiating modes will directly and dominantly contribute to the scattering field. As such, we focus on the radiating modes in the following. These modes can be decomposed into different components using the basis functions in Sec 2.3, and each component can be labeled by its angular momentum quantum number $〈 σ, l 〉$ , where $l$ and $σ$ are determined by Eq. (5) and Eq. (6), respectively. For band 1 (k > 0) in Fig. 6, the dominating components are $〈 + 1, - 1 〉, 〈 - 1, 0 〉$ and $〈 + 1, 0 〉$ . Therefore, under the excitation of an incident RH polarized plane wave ( $σ = - 1$ , consider the lower frequency excitation shown in Fig. 3(d)), the scattered light is mainly contributed by the partial wave channels with $m = σ + l = 0, - 1 and + 1,$ which coincides with the results in Fig. 4(e) and Fig. 5(e) ( $m = 0$ does not contribute to optical torque as can be seen from Eq. (3), while it contribute to light scattering). For band 2 (k > 0) in Fig. 6, the dominating components are $〈 + 1, + 1 〉, 〈 + 1, 0 〉$ and $〈 - 1, 0 〉$ . Therefore, under the excitation of an incident LH polarized plane wave ( $σ = + 1$ , consider the higher frequency excitation shown in Fig. 3(c)), the scattered light is mainly contributed by the partial wave channels with $m = σ + l = + 2, + 1 and - 1,$ which coincides with the results in Fig. 4(b) and Fig. 5(b).

Fig. 6 Band structure for an infinitely long periodic PEC helix. The lines denote the results of the analytical tape helix theory while the circles denote the full-wave results computed with COMSOL assuming a periodic array of helix on the xy plane with a large period of d = 4 μm. The gray area defines the light cone. The sold lines denote non-radiating guided modes while the dashed lines denote radiating modes.

Download Full Size | PDF

We note that such a selective excitation of chiral structures by chiral light is actually very general. The mirror symmetry breaking in presence of chiral structures (a nanohelix here) leads to different scattering/absorption of LH and RH polarized lights, which has also been widely investigated in bi-anisotropic media.

4. Summary

In summary, we studied the optical torque induced by the spin-orbit interaction of light in a metallic nanohelix structure. The conversion between the SAM and OAM can induce a broadband negative optical torque that rotates the helix in the opposite sense. We found that such a phenomenon can be attributed to the special eigen modes of the helix. The sign of the induced optical torque is determined by the polarization and the exciting frequency of the modes. It is also found that the light extinction can be quite independent of the torque. Our study shows that a simple metallic nanohelix has fruitful physics associated with light spin-orbit interaction and may find applications in the optical torque manipulations of chiral particles.

Funding

National Natural Science Foundation of China (NSFC) (11674204, 11404201); Hong Kong Research Grants Council (RGC) (AoE/P-02/12, HKBU209913); City University of Hong Kong (Project No. 9610388).

References

1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]

2. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

3. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]

4. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987). [CrossRef] [PubMed]

5. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

6. A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95(16), 168102 (2005). [CrossRef] [PubMed]

7. A. Kotnala and R. Gordon, “Quantification of high-efficiency trapping of nanoparticles in a double nanohole optical tweezer,” Nano Lett. 14(2), 853–856 (2014). [CrossRef] [PubMed]

8. P. C. Chaumet and A. Rahmani, “Dressed for success,” Nat. Nanotechnol. 9(4), 252–253 (2014). [CrossRef] [PubMed]

9. A. Rahimzadegan, M. Fruhnert, R. Alaee, I. Fernandez-Corbaton, and C. Rockstuhl, “Optical force and torque on dipolar dual chiral particles,” Phys. Rev. B 94(12), 125123 (2016). [CrossRef]

10. L. M. Zhou, K. W. Xiao, J. Chen, and N. Zhao, “Optical levitation of nanodiamonds by doughnut beams in vacuum,” Laser Photonics Rev. 11(2), 1600284 (2017). [CrossRef]

11. A. Rahimzadegan, R. Alaee, I. Fernandez-Corbaton, and C. Rockstuhl, “Fundamental limits of optical force and torque,” Phys. Rev. B 95(3), 035106 (2017). [CrossRef]

12. R. Shaniv and R. Ozeri, “Quantum lock-in force sensing using optical clock Doppler velocimetry,” Nat. Commun. 8, 14157 (2017). [CrossRef] [PubMed]

13. J. Li and P. M. Haney, “Optical spin transfer and spin-orbit torques in thin film ferromagnets,” Phys. Rev. B 96(5), 054447 (2017). [CrossRef] [PubMed]

14. M. G. Donato, E. Messina, A. Foti, T. J. Smart, P. H. Jones, M. A. Iatì, R. Saija, P. G. Gucciardi, and O. M. Maragò, “Optical trapping and optical force positioning of two-dimensional materials,” Nanoscale 10(3), 1245–1255 (2018). [CrossRef] [PubMed]

15. W. Li, N. Li, Y. Shen, Z. Fu, H. Su, and H. Hu, “Dynamic analysis and rotation experiment of an optical-trapped microsphere in air,” Appl. Opt. 57(4), 823–828 (2018). [CrossRef] [PubMed]

16. J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011). [CrossRef]

17. D. B. Ruffner and D. G. Grier, “Optical conveyors: A class of active tractor beams,” Phys. Rev. Lett. 109(16), 163903 (2012). [CrossRef] [PubMed]

18. A. Dogariu, S. Sukhov, and J. Saenz, “Optically induced ‘negative forces’,” Nat. Photonics 7(1), 24–27 (2013). [CrossRef]

19. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]

20. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998). [CrossRef]

21. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001). [CrossRef] [PubMed]

22. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002). [CrossRef] [PubMed]

23. S. L. Neale, M. P. MacDonald, K. Dholakia, and T. F. Krauss, “All-optical control of microfluidic components using form birefringence,” Nat. Mater. 4(7), 530–533 (2005). [CrossRef] [PubMed]

24. M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010). [CrossRef] [PubMed]

25. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010). [CrossRef] [PubMed]

26. Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4(1), 2374 (2013). [CrossRef] [PubMed]

27. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, 2003).

28. F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9(12), 776–778 (2015). [CrossRef]

29. K. Y. Bliokh, F. J. Rodriguez-Fortuno, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015). [CrossRef]

30. G. Li, M. Kang, S. Chen, S. Zhang, E. Y. B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13(9), 4148–4151 (2013). [CrossRef] [PubMed]

31. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef] [PubMed]

32. D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8(8), 610–614 (2014). [CrossRef]

33. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24(2), 430–443 (2007). [CrossRef] [PubMed]

34. J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 4(1), 6386 (2015). [CrossRef] [PubMed]

35. H. Magallanes and E. Brasselet, “Macroscopic direct observation of optical spin-dependent lateral forces and left-handed torques,” Nat. Photonics 12(8), 461–464 (2018). [CrossRef]

36. M. A. Ordal, R. J. Bell, R. W. Alexander Jr., L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985). [CrossRef] [PubMed]

37. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014). [CrossRef] [PubMed]

38. K. Ding, J. Ng, L. Zhou, and C. T. Chan, “Realization of optical pulling forces using chirality,” Phys. Rev. A 89(6), 063825 (2014). [CrossRef]

39. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009). [CrossRef] [PubMed]

40. R. N. Ji, S. W. Wang, X. X. Liu, H. J. Guo, and W. Lu, “Hybrid helix metamaterials for giant and ultrawide circular dichroism,” ACS Photonics 3(12), 2368–2374 (2016). [CrossRef]

41. M. Esposito, V. Tasco, F. Todisco, M. Cuscunà, A. Benedetti, M. Scuderi, G. Nicotra, and A. Passaseo, “Programmable extreme chirality in the visible by helix-shaped metamaterial platform,” Nano Lett. 16(9), 5823–5828 (2016). [CrossRef] [PubMed]

42. COMSOL is a finite-element-method electromagnetic solver, see “www.comsol.com

43. S. Sensiper, “Electromagnetic wave propagation on helical structures (a review and survey of recent progress),” Proc. IRE43, 149–161 (1955).

44. C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett. 105(24), 247401 (2010). [CrossRef] [PubMed]

45. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573–4588 (1995). [CrossRef] [PubMed]

46. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).

47. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University, 2002).

48. M. J. Berg, A. Chakrabarti, and C. M. Sorensen, “General derivation of the total electromagnetic cross sections for an arbitrary particle,” J. Quant. Spectrosc. Radiat. Transf. 110(1-2), 43–50 (2009). [CrossRef]

49. M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. M. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transf. 110(4-5), 323–327 (2009). [CrossRef]

50. J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(2), 026607 (2009). [CrossRef] [PubMed]

51. A. Dogariu and C. Schwartz, “Conservation of angular momentum of light in single scattering,” Opt. Express 14(18), 8425–8433 (2006). [CrossRef] [PubMed]

52. D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009). [CrossRef] [PubMed]

Mechanical effect of photonic spin-orbit interaction for a metallic nanohelix

Abstract

1. Introduction

2. Theory

2.1 Calculation of optical torque

2.2 Partial wave channel analysis of optical torque

2.3 Secular equation

3. Results and discussions

3.1. Partial wave channel analysis

3.2. Spin-orbit interaction

4. Summary

Funding

References

Cited By

Figures (6)

Equations (6)

Optics Express