![]() ![]() ![]() Controlled mechanical motions of microparticles in optical tweezers. Observation of a single-beam gradient force optical trap for dielectric particles. These studies will be of great help to understand the particle-laser trap interaction in various situations and promote exciting possibilities for exploring novel ways to control the mechanical dynamics of microscale particles.Īshkin A, Dziedzic JM, Bjorkholm JE, Chu S. Even in a simple pair of optical tweezers, the dielectric micro-sphere exhibits abundant phases of mechanical motions including acceleration, deceleration, and turning. The particle trajectory over time can demonstrate whether the particle can be successfully trapped into the optical tweezers center and reveal the subtle details of this trapping process. With the influence of viscosity force and torque taken into account, we numerically solve and analyze the dynamic process of a dielectric micro-sphere in optical tweezers on the basis of Newton mechanical equations under various conditions of initial positions and velocity vectors of the particle. In this paper, we utilize the ray optics method to calculate the optical force and optical torque of a micro-sphere in optical tweezers. In order to advance the flourishing applications for those achievements, it is necessary to make clear the three-dimensional dynamic process of micro-particles stepping into an optical field. When necessary to define a unique set of spherical coordinates for each point, the user must restrict the range, aka interval, of each coordinate.Known as laser trapping, optical tweezers, with nanometer accuracy and pico-newton precision, plays a pivotal role in single bio-molecule measurements and controllable motions of micro-machines. This article will use the ISO convention frequently encountered in physics, where the naming tuple gives the order as: radial distance, polar angle, azimuthal angle, or ( r, θ, φ ). (See graphic re the "physics convention"-not "mathematics convention".)īoth the use of symbols and the naming order of tuple coordinates differ among the several sources and disciplines. The depression angle is the negative of the elevation angle. The user may choose to ignore the inclination angle and use the elevation angle instead, which is measured upward between the reference plane and the radial line-i.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The polar angle may be called inclination angle, zenith angle, normal angle, or the colatitude. The radial distance from the fixed point of origin is also called the radius, or radial line, or radial coordinate. Nota bene: the physics convention is followed in this article (See both graphics re "physics convention" and re "mathematics convention"). Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3- tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. (See graphic re the "physics convention".) The azimuthal angle φ is measured between the orthogonal projection of the radial line r onto the reference x-y-plane-which is orthogonal to the z-axis and passes through the fixed point of origin- and either of the fixed x-axis or y-axis, both of which are orthogonal to the z-axis and to each other. The polar angle θ is measured between the z-axis and the radial line r. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a given point in space is specified by three numbers, ( r, θ, φ): the radial distance of the radial line r connecting the point to the fixed point of origin (which is located on a fixed polar axis, or zenith direction axis, or z-axis) the polar angle θ of the radial line r and the azimuthal angle φ of the radial line r. This is the convention followed in this article. ![]() Spherical coordinates ( r, θ, φ) as commonly used: ( ISO 80000-2:2019): radial distance r ( slant distance to origin), polar angle θ ( theta) (angle with respect to positive polar axis), and azimuthal angle φ ( phi) (angle of rotation from the initial meridian plane). 3-dimensional coordinate system The physics convention. ![]()
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