Local angular momentum induced dual orbital effect

As a fundamental property of light, the angular momentum (AM) has been of great interest. Generally, the right- and left-handed circularly polarized light can carry a spin angular momentum (SAM) of ±ℏ per photon, while the linearly polarized light carries no SAM per photon. An optical field with an azimuthal angular dependence of exp(ilϕ), where l is an integer, carries an orbital angular momentum (OAM) of lℏ per photon.^1,2 In general, SAM and OAM can be experimentally distinguished according to their different mechanical actions on probing microparticles. SAM usually makes the microparticle rotate around its axis, while OAM causes orbital motion of the microparticles around the beam axis.^3,4 The interaction between SAM and OAM has led to the discovery of new phenomena and opened a route to many promising applications.^5–9 

In an inhomogeneous anisotropic medium, SAM of light is converted into OAM (i.e., spin-to-orbit conversion, STOC).^10–12 In the photonic crystal slab, the STOC has been achieved by an optical vortex generated for momentum-space polarization vortices centered at bound states in the continuum.¹³ In a homogeneous and isotropic medium, the STOC is also possible. A tightly focused circularly polarized light partly transfers its SAM into OAM, which leads to the generation of a helical phase in the longitudinal component of the electric field.^14–16 In some situations, however, even if light beams carry no OAM and without STOC, the orbital motion of probing particles around the beam axis can also be observed in an optical trap.^17–22 This phenomenon is that the spin part of the internal energy flow causes the orbital motion. We have presented and demonstrated novel OAMs related to the vector optical fields, caused by the azimuthally varying elliptical polarization state²³ and the curl of polarization.²⁴ Very recently, by using a circularly polarized light with linearly varying radial phase, we have also achieved the control of local AM density via the STOC caused by the radial intensity gradient, in the planes away from the focal plane as the asymmetric intensity distribution existed there.²⁵ Such a spin-dependent local AM density can induce a counterintuitive orbital motion of isotropic microparticles in optical tweezers. However, such an orbital motion cannot be observed in the focal plane, where the isotropic probing particles are trapped at the strongest ring and will not feel the net AM because the intensity across the strongest ring exhibits a symmetric distribution along the radial direction. In fact, such an orbital motion of the isotropic probing particles cannot also be observed by the Laguerre–Gaussian (LG) beams in any plane due to the same reason as above. To detect the local AM density caused by the radial intensity gradient through observing the orbital motion of probing particles, a certain symmetry breaking is needed. In Ref. 25, the symmetry of the radial intensity distribution about the strongest ring is broken. Another possible solution for introducing symmetry breaking is to select symmetry-broken particles (e.g., geometric or optical properties).

Here, we demonstrate a dual orbital effect caused by a combination of the asymmetric Janus particles and the local AM density via the radial intensity gradient. A circularly polarized light with a linearly varying radial phase, in the absence of intrinsic OAM, is focused on an annular profile, which stably traps the Janus particles moving along completely opposite orbital directions at different radial locations (both sides of the strongest ring) due to the nonzero net local AM. The manipulation of the local AM and the observed novel dual orbital effect in the absence of intrinsic OAM provide new insight into understanding the light–matter interaction, paving the way to some new applications involving the sorting and delivery of microparticles.

The z component of the local AM density of a circularly polarized light without helical phase can be expressed as^3,25,26

where u is the complex amplitude of an optical field, σ = ±1 stand for the right- and left-handed circularly polarized states, ω is the angular frequency of light, and ɛ₀ is the permittivity of vacuum.

The spin-dependent AM density can be manipulated by the radial intensity gradient term of ∂|u|²/∂r in Eq. (1), and this is easily achieved with a Gaussian beam. However, since a Gaussian beam has the strongest intensity at the center and its intensity decreases gradually as it is away from the center, as a result, its radial intensity gradient can only be unidirectional, which leads to a unidirectional local AM density j_z. In order to achieve the bidirectional local AM density, it needs to generate an annular beam whose strongest intensity is not at the center of the beam but exhibits as the strongest ring. Although higher order LG beams can also meet this requirement, it is difficult to clearly observe the spin-dependent local AM due to the presence of the intrinsic OAM. For this reason, a linearly varying radial phase is introduced into the input optical field, which results in a substantial change in the spatial structure of the optical field, i.e., a fundamental Gaussian beam will become an annular optical field without intrinsic OAM.²⁵ The annular optical field has the opposite intensity gradients on both sides of the strongest ring, which makes it possible to realize a bidirectional local AM density j_z.

The annular optical field at the focal plane can be generated by modulating the input field at the pupil plane with a linearly varying radial phase of exp(−ik_rr), where k_r = 2nπ/r₀, r and r₀ are the radial coordinate and the pupil (beam) radius, respectively, and n determines the radial wave vector k_r. The linearly varying radial phase is easily realized by a spatial light modulator (SLM).²⁷ Thus, the input field can be written as

After a lens with a focal length of f, the input field is focused to form an annular optical field in the focal plane, as shown in Fig. 1(a). Such an annular optical field has a Gaussian-like intensity distribution of $exp[(r−Rm)2/R02]$ along the radial direction (R₀ is the half width of the annulus), which is symmetrical about the strongest ring with a radius of R_m, and it is quite different from the LG beams because it carries no OAM. The annular optical field has R_m = k_rf/k = nλf/r₀ and R₀ = 2f/kr₀ = fλ/πr₀ (λ is the wavelength of light). When calculating the intensity pattern at the focal plane in Fig. 1(a), we use the Fourier transform method to deal with the Wolf–Richards vectorial integral²⁸ to speed up the calculation,²⁹ and then, j_z can be calculated by Eq. (1).

Let us take the left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) light as examples, Figs. 1(b) and 1(e) show the intensity patterns of the focused LCP and RCP lights with a linearly varying radial phase (n = 7), it can be seen that they have the same annular profile. Figures 1(c) and 1(f) show the corresponding local AM density distributions for the LCP and RCP lights, respectively; obviously, the AM densities have the bidirectional characteristic. Figures 1(d) and 1(g) correspond to the partly enlarged views of Figs. 1(c) and 1(f), respectively, where the arrows represent the directions of the azimuthal momentum densities p_ϕ. When comparing the LCP light with the RCP light, we can find that the directions of the local AM density will be reversed by changing the sign of SAM, implying that we can achieve a bidirectional AM density in the annular optical field.

Figures 2(a1) and 2(a2) show the intensity profiles and the local AM densities of an RCP annular optical field along its diameter parallel to the x axis in the focal plane. We can see that they have the bidirectional local AM densities, and they have indeed equal magnitude and opposite signs on both sides of the strongest ring. In order to probe the local effect of the spin-dependent AM density in the experiment, the probing particles need to be selected carefully. That is to say, the center of the probing particle should deviate from the strongest ring (with a radius of R_m) to obtain a nonzero local AM (J_z) acting on the probing particle, which can be expressed by $Jz∝∫Rm+Δr−RRm+Δr+Rjzdr$⁠, where R is the radius of the probing particle and Δr is the distance between the particle center and the strongest ring. However, Δr can be positive or negative, which implies that the particle is trapped in the outer or inner side of the strongest ring.

In general, a dielectric (e.g., SiO₂) or polymer (polystyrene) particle will be stably trapped at the strongest location of an optical field, however, which would make it impossible to observe the local effect of the spin-dependent AM in the focal plane because the total AM J_z exerted on the particle is zero when Δr = 0, and hence, there is no orbital motion of the probing particle along the annulus.

Janus particles, a kind of asymmetric particles, are named by the two-faced Roman god Janus, which have two distinct sides with different surface features, structures, or/and compositions. This asymmetric structure enables the combination of different or even incompatible physical effects³⁰ and has found applications in light-driven rotation and optical manipulation.^31,32 For the Janus particles we used, its matrix is a SiO₂ sphere with a radius of R = 1.5 μm, and half of its spherical surface has a gold coating with a thickness of h = 20 nm, as shown in Figs. 2(b1) and 2(b2). Here, we present a theoretical study of the optical force and torque exerted on a micrometer-sized Janus particle. Furthermore, to clarify the rotational behavior, we use two rotational degrees of freedom α and γ, which represent the rotation angles of the Janus particle around the x₀- and z₀-axes, respectively. The three-dimensional illustration in Fig. 2(b1) corresponds to the condition of α = 0 and γ = 0, in which the interface between the gold-coated and non-coated halves is located in the x₀–z₀ plane.

The numerical calculations show that the center of the Janus particle will be stably trapped in a radial location slightly away from the strongest ring, and the non-coated hemisphere of the Janus particle always points to the strongest ring (see the supplementary material for details), which is in agreement with the well-known fact that the dielectric particle is favorite to be trapped in the stronger intensity location while the metal particle is favorite to be trapped in the dark location.^3,26 Therefore, for the Janus particle trapped in the annular optical field, there will be two equilibrium radial locations on the inner and outer sides of the strongest ring, respectively. As shown in Figs. 2(c1) and 2(c2), θ is the angle formed by the interface normal vector v (which is perpendicular to the two-hemisphere interface of the Janus particle) and the x axis. ϕ is the azimuth angle with respect to the x axis in the coordinate system of the annular optical field. For the two equilibrium locations, the pose angles are completely different. Our calculation shows that the two equilibrium locations have distances of Δr₁ = −0.36 μm and Δr₂ = 0.36 μm away from the strongest ring (with R_m = 7.3 μm), respectively. The Janus particles in different equilibrium locations are subjected to different net AMs, which have opposite directions. As a result, the probing Janus particles trapped in the inner and outer sides of the strongest ring move along the clockwise or anticlockwise directions (or opposite case, dependent on SAM), respectively.

If the two hemispherical surfaces of a particle have the same dielectric nature but different refractive indices, the particle trapped in the annular optical field can also have, in principle, two equilibrium radial locations, but the interval between the two equilibrium locations is very small. Therefore, it is very hard to observe the dual orbital effect.

In order to verify the dual orbiting motion of the Janus particles, we established optical tweezers with an annular beam (as generated above) in an inverted microscope system, as shown in Fig. 3(a). A continuous wave laser (Verdi-5, Coherent Inc, λ = 532 nm, maximum power of 5 W) is expanded by a telescope system (L1 and L2) to produce an approximate top-hat profile, then it is incident on a phase-only SLM (Holoeye PLUTO-2-VIS-016, with 1920 × 1080 pixels), which locates at the input plane of a 4f system composed of a pair of lenses (L3 and L4, f = 300 mm). The 4f system is used to transfer the image plane on SLM to the rear focal plane of the objective lens (OL). A half wave plate (HWP1) is used to adjust the polarization state to be horizontally polarized, and a designed holographic grating is used to generate the annular optical field.²⁵ 

In the Fourier plane of L3, we used a spatial filter (SF) to pick up the +first-order diffracted (from the SLM), which gets the linearly varying radial phase of exp(−ik_rr), and then, it is collimated by L4 to enter into the inverted microscope system. The combination of a half wave plate (HWP2) and a PBS is used to adjust the light intensity, and a quarter wave plate (QWP) is used to convert the linear polarization into circular polarization. After focusing by an OL (NA = 0.65), the circularly polarized light with the phase of exp(−ik_rr) is turned into an annular optical field in the focal plane (as illustrated in Fig. 1). The Janus particles are dispersed in deionized water solution sandwiched between two glass coverslips separated by a spacer with a thickness of ∼100 μm. The orbital motion of the Janus particle is observed and recorded by a charge coupled device camera (CCD) (with a resolution of 1280 × 1024 pixels and a maximum frame rate of 60 fps) with the help of white light illumination from a light-emitting diode (LED). A cut-off filter (F) is inserted in the front of the camera to filter out the green light.

Due to the strong scattering force of light, the Janus particle is pushed to move along the light propagation direction (z) in the deionized water solution until it contacts the top cover slip to form a longitudinal confinement. Therefore, the particle will no longer move in the z direction, and we need to only consider the motion in the transverse plane.

In Fig. 3(a), when the LCP light enters into the OL, two stable trapping equilibrium states are observed, i.e., the Janus particles are trapped in the inner and outer sides of the strongest ring, and the directions of the orbital motion are clockwise and anti-clockwise, respectively. The particles move on two orbits with the radii of r₁ ≈ 6.9 μm and r₂ ≈ 7.7 μm, respectively, as shown in Fig. 3(b). The difference between r₁ (r₂) and the strongest ring [with R_m ≈ 7.3 μm, shown by the black circle in Fig. 3(b)] is Δr₁ ≈ −0.4 μm (Δr₂ ≈ 0.4 μm), which agrees with the theoretical calculation (Δr₁ = −0.36 μm and Δr₂ = 0.36 μm). Figure 3(c) shows the dependence of the pose angle θ on the azimuth angle ϕ, clearly θ = π + ϕ for the anticlockwise orbiting (r₂), while θ = ϕ for the clockwise orbiting (r₁). These experimental results further confirmed our analysis about the attitude of the trapped Janus particles, that is, whether the Janus particle is trapped at the outer side or the inner side of the strongest ring of the annular optical field, the non-coated hemisphere always points to the strongest ring. This conclusion is also experimentally supported by the fact that when the particle is trapped in the outer (inner) orbit with a radius of r₂ (r₁), the bright hemisphere always points toward the strongest ring. For the gold-coated film, the reflection of light is more than the transmission, so the gold-coated hemisphere is darker while the non-coated hemisphere is brighter when illuminated by the white light LED. Figure 3(d) shows a series of snapshots at different moments, and the dull blue (outer side) and the dull red (inner side) circles represent the orbits of r₂ and r₁, respectively. Obviously, the experimental results agree with that shown in Fig. 3(c) very well.

Subsequently, the experimental results of the RCP light are shown in Fig. 4. When comparing Fig. 4(a) with Fig. 3(b), it is clear that the orbital motions for the inner and outer sides are completely opposite, which implies that the direction of the local AM density is also opposite for the RCP and LCP lights. The trapped Janus particles also move along the inner and outer orbits with the radius of r₁ and r₂, respectively. The directions of the orbital motion are anticlockwise and clockwise, respectively. As for the motion attitudes of the trapped particles, Fig. 4(b) shows the dependence of the pose angle θ on the azimuth angle ϕ, as defined in Fig. 2(c); clearly, it meets θ = π + ϕ and θ = ϕ for the clockwise (r₂) and the anticlockwise (r₁) orbiting, respectively. In addition, the experimental results shown in Fig. 4(c) reveal that the non-coated hemisphere always points to the strongest ring as the particle orbiting regardless of clockwise or anticlockwise, which is exactly as we expected.

It is worth noting that the observed dual orbital motion of Janus particles is dominated by the local SAM density, while the contribution of the STOC is negligible. The STOC is related to the longitudinal field only; however, in our experiment, the longitudinal field intensity is only about 2% of the total field intensity. On the other hand, for the circularly polarized light, the STOC can only cause the unidirectional orbital motion of the trapped particles, while it is impossible to produce the dual orbital effect in the opposite directions.

Figure 5 shows the relationship between the mean angular velocity of the orbital motion and the input laser powers under the condition of input LCP light, and the fluctuation of angular velocity is given by the standard deviation. The sign (±) of the angular velocity in Fig. 5 represents the direction of the orbital motion, which is negative for the inner orbital motion [clockwise, as shown in Fig. 3(b)]. It can be seen that the absolute mean angular velocity gradually increases as the laser power increases because the laser energy affects the local AM density by ∂|u|²/∂r in Eq. (1). At the same time, we can see that the fluctuation becomes a little larger as the laser power increases, which indicates that the stronger thermal effect may aggravate the instability of the equilibrium state.

Based on the manipulation of the local AM density in the focal field and the use of asymmetric probing particles, i.e., Janus particles, we have demonstrated the dual orbital motion of the probing particle in the absence of intrinsic OAM. A circularly polarized light with a linearly varying radial phase is focused to form an annular optical field in the focal plane and achieve a local distributed AM density, which possesses the opposite directions in both sides of the strongest ring. Stable optical trapping of the Janus particle in both sides of the strongest ring is realized in the annular optical field. As a result, dual orbital motions of the Janus particle with the opposite directions in both sides of the strongest ring are observed. In the process of orbital motion, the pose angle of the Janus particle changes continually so that the non-coated hemisphere of the Janus particle always points to the strongest ring. As far as we know, this is the first experimental observation of the so-called “dual orbital effect,” i.e., the particle trapped in a focused optical field moves orbitally with completely opposite directions at different radial locations due to the spin-dependence local AM density instead of intrinsic OAM. The findings provide new insights into the physics underlying the light–matter interaction, paving the way for some new applications involving the sorting and delivery of microparticles.

See the supplementary material for additional details.

This work was supported by the National Key R&D Program of China (Grant Nos. 2017YFA0303800 and 2017YFA0303700) and the National Natural Science Foundation of China (Grant Nos. 12074196, 11774183, and 12074197).

The authors would like to thank support of the Collaborative Innovation Center of Extreme Optics.

The authors declare no conflicts of interest.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.