We demonstrate both theoretically and experimentally that gradients in the phase of a light field exert forces on illuminated objects, including forces transverse to the direction of propagation. This effect generalizes the notion of the photon orbital angular momentum carried by helical beams of light. We further demonstrate that these forces generally violate conservation of energy, and briefly discuss some ramifications of their non-conservativity.Light's ability to exert forces has been recognized since Kepler's De Cometis of 1619 described the deflection of comet tails by the sun's rays. Maxwell demonstrated that the momentum flux in a beam of light is proportional to the intensity and can be transferred to illuminated objects, resulting in radiation pressure that pushes objects along the direction of propagation. This axial force has been distinguished from the transverse torque exerted by helical beams of light carrying orbital angular momentum (OAM) [1]. We demonstrate theoretically and confirm experimentally that OAM-induced torque is a special case of a general class of forces arising from phase gradients in beams of light. We also demonstrate that phase-gradient forces are generically non-conservative, and combine them with the conservative forces exerted by intensity gradients to create novel optical traps with structured force profiles.Our experimental demonstrations of phase-gradient forces make use of extended optical traps created through shape-phase holography [2,3,4] in an optimized [5] holographic optical trapping [6,7] system. Holographically sculpted intensity gradients enable these generalized optical tweezers [8] to confine micrometer-scale colloidal particles to one-dimensional curves embedded in three dimensions. Independent control over the intensity and phase profiles along the curve then provide an ideal model system for characterizing the forces generated by phase gradients in beams of light. OPTICAL FORCES DUE TO PHASE GRADIENTSThe vector potential describing a beam of light of frequency ω and polarizationε may be written in the form A(r, t) = A(r) exp (i ωt)ε.(1)Assuming uniform polarization (and therefore a form of the paraxial approximation), the spatial dependence,is characterized by a non-negative real-valued amplitude, u(r), and a real-valued phase Φ(r). For a plane wave propagating in theẑ direction, Φ(r) = −kz, where k = n m ω/c is the light's wavenumber, c is the speed of light in vacuum, and n m is the refractive index of the medium. Imposing a transverse phase profile ϕ(r) on the wavefronts of such a beam yields the more general formwhereẑ · ∇ϕ = 0 and where the the direction of the wavevector,now varies with position. The associated electric and magnetic fields are given in the Lorenz gauge by E(r, t) = − ∂A(r, t) ∂t and (5)where µ is the magnetic permeability of the medium, which we assume to be linear and isotropic. Following the commonly accepted Abraham formulation [9], the momentum flux carried by the beam iswhere I(r) = u 2 (r) is the light's intensity, and where w...

We demonstrate both experimentally and theoretically that a colloidal sphere trapped in a static optical tweezer does not come to equilibrium, but rather reaches a steady state in which its probability flux traces out a toroidal vortex. This non-equilibrium behavior can be ascribed to a subtle bias of thermal fluctuations by non-conservative optical forces. The circulating sphere therefore acts as a Brownian motor. We briefly discuss ramifications of this effect for studies in which optical tweezers have been treated as potential energy wells.Most discussions of the dynamics of optically trapped particles assume at least implicitly that the forces exerted by an optical tweezer [1] are path-independent and therefore conserve mechanical energy. Optical forces due to gradients in the intensity are manifestly conservative in this sense [2]. Radiation pressure, by contrast, is not [2,3]. The experimental studies described in this Letter demonstrate that the non-conservative component of the optical force has measurable consequences for the dynamics of optically trapped colloidal spheres. In particular, the probability density for a sphere trapped in a static optical tweezer exhibits steady-state toroidal currents, a phenomenon we call a fountain of probability. We use the Fokker-Planck formalism to explain how nonconservative forces bias random thermal fluctuations to induce circulating probability currents. Figure 1 schematically represents the deceptively simple system. A single colloidal sphere is drawn to the focus of a converging laser beam by forces arising from gradients in the beam's intensity [1,2]. These intensitygradient forces establish a three-dimensional potential energy well, V (r), determined by the local intensity, I(r). A particle trapped in this well also experiences radiation pressure that drives it downstream with a force proportional to I(r). In the absence of thermal fluctuations, a trapped particle would come to rest at a stable mechanical equilibrium downstream of the focus.Treating the displaced equilibrium point as the origin of an effective potential energy well is tempting but misleading. To appreciate the problem, consider a thermally driven trajectory such as the example shown schematically in Fig. 1. Were the system in thermodynamic equilibrium, forward and reverse trajectories around this loop would have equal probability. Because the light is more intense near the optical axis, however, radiation pressure biases the random walk in the forward direction. This departure from detailed balance should induce irreversible circulation in the particle's otherwise random fluctuations [4].We demonstrate this effect by observing the motions of colloidal silica spheres 2.2 µm in diameter (Bangs Lab-

Mechanical equilibrium at zero temperature does not necessarily imply thermodynamic equilibrium at finite temperature for a particle confined by a static but nonconservative force field. Instead, the diffusing particle can enter into a steady state characterized by toroidal circulation in the probability flux, which we call a Brownian vortex. The circulatory bias in the particle's thermally driven trajectory is not simply a deterministic response to the solenoidal component of the force but rather reflects interplay between advection and diffusion in which thermal fluctuations extract work from the nonconservative force field. As an example of this previously unrecognized class of stochastic heat engines, we consider a colloidal sphere diffusing in a conventional optical tweezer. We demonstrate both theoretically and experimentally that nonconservative optical forces bias the particle's fluctuations into toroidal vortexes whose circulation can reverse direction with temperature or laser power.

The video stream captured by an in-line holographic microscope can be analyzed on a frame-by-frame basis to track individual colloidal particles' three-dimensional motions with nanometer resolution, and simultaneously to measure their sizes and refractive indexes. Through a combination of hardware acceleration and software optimization, this analysis can be carried out in near real time with off-the-shelf instrumentation. An efficient particle identification algorithm automates initial position estimation with sufficient accuracy to enable unattended holographic tracking and characterization. This technique's resolution for particle size is fine enough to detect molecular-scale coatings on the surfaces of colloidal spheres, without requiring staining or fluorescent labeling. We demonstrate this approach to label-free holographic flow cytometry by detecting the binding of avidin to biotinylated polystyrene spheres.

The core performance elements of global navigation satellite system include availability, continuity, integrity and accuracy, all of which are particularly important for the developing BeiDou global navigation satellite system (BDS-3). This paper describes the basic performance of BDS-3 and suggests some methods to improve the positioning, navigation and timing (PNT) service. The precision of the BDS-3 post-processing orbit can reach centimeter level, the average satellite clock offset uncertainty of 18 medium circular orbit satellites is 1.55 ns and the average signal-inspace ranging error is approximately 0.474 m. The future possible improvements for the BeiDou navigation system are also discussed. It is suggested to increase the orbital inclination of the inclined geostationary orbit (IGSO) satellites to improve the PNT service in the Arctic region. The IGSO satellite can perform part of the geostationary orbit (GEO) satellite's functions to solve the southern occlusion problem of the GEO satellite service in the northern hemisphere (namely the "south wall effect"). The space-borne inertial navigation system could be used to realize continuous orbit determination during satellite maneuver. In addition, high-accuracy space-borne hydrogen clock or cesium clock can be used to maintain the time system in the autonomous navigation mode, and stability of spatial datum. Furthermore, the ionospheric delay correction model of BDS-3 for all signals should be unified to avoid user confusion and improve positioning accuracy. Finally, to overcome the vulnerability of satellite navigation system, the comprehensive and resilient PNT infrastructures are proposed for the future seamless PNT services.