Central force and scattering problems
Answers
Central Force Problems :
Central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.
The solution of this problem is important to classical mechanics, since many naturally occurring forces are central. Examples include gravity and electromagnetism as described by Newton's law of universal gravitation and Coulomb's law, respectively. The problem is also important because some more complicated problems in classical physics (such as the two-body problem with forces along the line connecting the two bodies) can be reduced to a central-force problem. Finally, the solution to the central-force problem often makes a good initial approximation of the true motion, as in calculating the motion of the planets in the Solar System.
Scattering or Gradient force Problems :
Division of optical force into gradient force and scattering force is a decades long problem. The concepts of “gradient force” and “scattering force” are very commonly applied in the interpretation of optical trapping simulation and experiment, but no one has seen their true face except for a few special cases. Such a division is important because the gradient force is responsible for optical trapping, while scattering force is responsible for particle transportation. All previous studies either has unjustified approximations or consider particle small (dipole) or large (geometrical optics) compare to the wavelength of light. Here, we present a rigorous, accurate, and efficient method to separate the gradient force and the scattering force using Fourier Transformation. This approach can, in principles, handle spherical particles of arbitrary size and composition, as long as the total optical force can be computed by, for example, Mie scattering theory plus Maxwell stress tensor. Particle of different sizes and compositions trapped by an optical tweezers are considered, which includes dipolar/Mie sized particle made up of dielectric or metal. Not surprisingly, the numerical aperture and aberration also play an important role. We fully tested the accuracy and stability of this approach: it is sufficiently accurate as long as a sufficiently large unit cell is employed such that the optical force in the cell boundary can be treated as zero.