Spin-to-Orbital Angular Momentum Conversion via Light Intensity Gradient

Besides a linear momentum, optical fields carry an angular momentum (AM), which have two intrinsic components: one is spin angular momentum (SAM) related to the polarization state of the field, and the other is orbital angular momentum (OAM) caused by the helical phase due to the existence of topological azimuthal charge. The two AM components of the optical field may not be independent of each other, and the Spin-to-Orbital AM conversion (STOC) under focusing will create a spin-dependent optical vortex in the longitudinal filed. Here we demonstrate a new mechanism (or novel way, new way, specific process) for the STOC based on a radial intensity gradient. The radial phase provides an effective way to control the local AM density, which induce counterintuitive orbital motion of isotropic particles in optical tweezers without intrinsic OAM. Our work not only provides fundamental insights into the spin-orbit interaction of light, but also push towards possible applications in optical micro-manipulation.


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Angular momentum (AM) is one of important characteristics of light 1,2 and has attracted increasing attention in a variety of applications, such as optical manipulations 3,4 , quantum information [5][6][7][8][9] , optical communications 10,11 and imaging [12][13][14] . For paraxial fields in free space, AM can be formally separated into a spin angular momentum (SAM) associated with the right-or left-hand circular polarization corresponding to positive or negative helicity of    or   and an orbital angular momentum (OAM) caused by a helical phase of exp(jm), where m is the azimuthal topological charge and  is the azimuthal angle. In general, SAM and OAM can be experimentally distinguished according to their different mechanical actions on microparticles, because SAM usually makes the particles rotate around its own axis, while OAM causes orbital motion of the particles around the beam axis. This phenomenon is a conventional characterization method and distinguishing rule for SAM and OAM in experiment. However, in extreme cases, they cannot be separated. It has been known that Spin-to-orbit AM conversion (STOC) could occur when a circularly polarized Gaussian beam is converged by a lens 16,17 , generating a spin-dependent optical vortex in the longitudinal field component. In fact, this conversion process verified in experiment depends mainly on the mechanical property of spin-dependent local AM (ref. 17), because the microparticles interact with the part of optical field.
For the circularly polarized optical field carrying the helical phase, the AM density in the direction of propagation is given by 15 22 where u is a complex scalar function describing the distribution of the field amplitude, which satisfies the wave equation under the paraxial approximation. Obviously, in equation (1), the first term originates from the contribution of intrinsic OAM caused by the topological defect introduced by the helical phase in the field; while the second term arises from the combination of SAM () and the radial intensity gradient (RIG).
We usually change the azimuthal topological charge m to effectively control the AM carried by the beam.
In fact, changing the RIG may also provide an effective way to manipulate the sign and value of the local AM density and it may help us to deeply understand the STOC. But an analysis of the relation between the local AM density, the STOC and the RIG is still missing and remains a problem 16 .
Here we provide an effective way to manipulate the RIG and harness local AM density. We predict in theory and validate in experiment a spin-dependent local AM associated with the RIG. Such a spindependent local AM can induce a counterintuitive orbital motion of isotropic particles in optical tweezers and indicate there is a new STOC mechanism. This breaks the cognition that STOC will create a spindependent optical vortex, in which the RIG plays an important role in the process.

THEORETICAL ANALYSIS
To detect or characterize the AM experimentally, the optical trap or tweezers provides an effective method by the aid of exchange of mechanical torque acting on microparticles based on the transfer of AM. In previous works, the intensity gradient had been ignored because its symmetric distribution near the focus.
For the circularly polarized Gaussian beam, the microparticle is usually trapped in the beam center results zero orbital rotation torque because the opposite RIG across the center. Even for the circularly polarized Laguerre-Gaussian beam, the probed dielectric microparticles will be stably trapped at the strongest ring with a defined radius, but the azimuthal force from the near odd-symmetric RIG is also zero. This implies the fact that the local AM density caused by the non-symmetric RIG will have contribution to the orbital motion of the spherical microparticle trapped at the strongest ring, but a key problem is how to achieve the non-symmetric RIG (or the non-odd-symmetric RIG) about the strongest ring.
The radial phase (RP) may provide an effective way to control the RIG. The RP can be classified into two categories: Category-(i), which contains the odd power of radial coordinate only, and Category-(ii), which contains the even power of radial coordinate only. Category-(i) is completely different from Category-(ii) that the odd (even) power of radial coordinate will (will not) dramatically change the spatial structure of the optical field. When the RP contains the even power of radial coordinate only 18,19 , a spin-3/2 light has been predicted 18 . Here we devote to harness the local AM density based on the RIG, by using the linearly-varying RP without the intrinsic OAM caused by the topological defect of helical phase, unlike Ref. 17 requires the tightly focused circularly polarized vortex. Since the introduction of the linearly-varying (odd power) RP results in the substantial change of spatial structure of the optical field, for instance, a fundamental Gaussian beam will become a doughnut-like pattern.
Under the paraxial approximation, the transversal electric field of light with linearly-varying RP without the helical phase in free space can be written as follows where A(r) represents the amplitude of the optical field, q is called the radial index that can be an any number 1(d3)] is opposite to that of q > 0. This is completely different from the Laguerre-Gaussian field, which always exhibits a near symmetric intensity distribution with respect to the strongest ring, in any plane along the propagation direction. Clearly, the linearly-varying RP provides an effective method to control the RIG.
If the size of microparticles is comparable with the width of the ring of the focused optical field with linearly-varying RP, we consider the integral of the AM density jz across the ring (from RP a to RP + a) to describe the local interaction between the focused field and microparticles The gradient force constrains the particles to the strongest ring and a is the radius of the probing particle. As shown in Fig. 1, in the Fourier plane of the lens, the local doughnut-like pattern has a Gaussian-like intensity distribution as exp[(rRP) 2 /a 2 ], so the net local AM L z J will be near zero, which cannot result in the orbital motion of microparticles. In the plane z ≠ 0, however, L z J will be nonzero due to the asymmetric radial intensity (i.e., non-odd symmetric RIG) about r  RP, meaning that the trapped microparticles will move along a circular orbit. We have simulated the radial-varying intensity and AM density in the vicinity of the is odd symmetric about r = RP, so the spin flow is also odd symmetric about r = RP. Therefore, the macroscopic spin flow vanishes, this is the reason why a trapped calcite particle only spins about own axis while has no orbital motion, which is similar to the circular polarized Laguerre-Gaussian beam 23 . In contrast, in the z ≠ 0 plane, the optical field becomes asymmetric in intensity about r = RP, as shown by the asymmetric dark green doughnut-like in Fig. 2(b); this is, the radial intensity gradient lacks the odd symmetry about r = RP, so the spin flow has also no odd symmetry about r = RP. As a result, the macroscopic spin flow becomes nonzero along the azimuth, as shown by open red arrow in Fig. 2(b), this is the reason why the trapped particles have orbital motion.

EXPERIMENTAL RESULTS
To confirm the above theoretical analysis and simulation results, we experimentally generate the optical  Fig. 4(c), the trapped particles have increased to ten and move clockwise for the right-circularly polarized light; while when switching to q  7 in Fig. 4(d), the motion direction of trapped particles is synchronously reversed. Obviously, the orbital motion of the trapped particles arises from the transfer of the local AM L z J to particles through the RIG, which can be effectively controlled by the RP.
In addition, L z J has also an important feature that is its longitudinal dependence. The trapped particles move clockwise for q  +6 and σ  +1 in the case of z < 0 in Fig. 5(a), while the sense of the orbital motion of the trapped particles is reversed for q  +9 and σ  +1 in the case of z > 0 in Fig. 5(b). More importantly, the local AM caused by the RIG can have the same or opposite sign as the intrinsic OAM originated from the vortex phase (see Fig. S5 for details). We can load the hologram with the linearly-varying RP of q  +6 and the vortex phase of m  +1 on SLM simultaneously, then the focused field exhibits still the doughnut shape. When σ  +1, the trapped particles almost stop behind the geometric focal plane (z > 0), the local AM from the linearly-varying RP of is compensated with the intrinsic OAM of helical phase, which means that the net total local AM acting on particles is almost zero in Fig. 6(a). Switching the polarization from σ  +1 to σ  1, the trapped particles move clockwise with a shorter orbital period of ~8 s in Fig. 6(b).

Discussion and Conclusion
Page 7 of 17 We experimentally demonstrate the fact that in the absence of intrinsic OAM, the net nonzero local AM density may not only be different in magnitude but also has the different sign from the SAM with the linearlyvarying RP. The sense and velocity of orbital motion of microparticles depend on SAM and the RIG and this local AM can be continuously changed by selecting arbitrary radial index q. The effective torque on particles from the local L z J is equivalent to that from the intrinsic OAM, and in this sense the radial phase provides a new way for optical manipulation. Our scheme with continuously control of the local AM density based on the linearly-varying RP is also compatible with other degree of freedom of light. The observed orbital motion of the isotropic particles in optical tweezers using the optical fields without intrinsic OAM may show the same the experiment phenomena as the OAM caused by the spin-to-orbital conversion from a tightly focused circularly polarized Gaussian beam 17 and an inhomogeneous and anisotropic metamaterial 26 . By numerical simulation, the intensity of the longitudinal field is very small comparing with transverse fields (see Fig. S6) and it indicates the RIG plays an important role in the STOC.
In conclusion, we have theoretically predicted and verified the STOC associated with the RIG through optical trapping experiments. The result breaks the limitation that orbital motion must be associated with the azimuthal phase gradient and enriches the category of AM. This spin-dependent AM will also give us a deeper understanding of SAM in optics as well as conversion of spin to orbit. Since this AM can be easily generated and arbitrarily tunable, it opens a novel route to control the light-matter interaction and optical metrology and it may show great applications in optical micro-manipulation and micro-fabrication. Besides, it is also expected to be found in other natural waves such as electron beams and acoustic waves [27][28][29] .
particles are observed another objective O2 (100× and NA = 0.95). Two charge coupled device cameras CCD1 and CCD2 with resolution of 1280×1024 pixels and maximal frame rate of 60 fps are used to record the manipulation process of the trapped particles.
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