An Equilateral Triangle Waveguide Beam Splitter
In this paper an optical beam splitter based on an equilateral triangle waveguide (ETW) is studied theoretically and numerically. We show that an optical beam splitter bases on ETW formed only when the length of the waveguide and the location of incident light are appropriate. When the length of the ETW is one third of the self-imaging length, the incident light is divided into three identical and symmetrical beams; while the length of the ETW is one nine of the self-imaging length, the incident light is divided into twenty- seven identical and symmetrical beams, if the number is less than twenty-seven, that is because some of images overlap with each other. It is expected that the results obtained here will help to design a new splitter.
Resonant mode lifetimes due to boundary wave emission in equilateral triangular dielectric cavities
Lifetimes are estimated for the two-dimensional resonant optical modes of a dielectric cavity with an equilateral triangular cross section, that are approximately confined by total internal reflection. Exact solutions of a two-dimensional scalar wave equation for triangular geometry with Dirichlet boundary conditions are used to describe approximately the vector fields of the possible transverse electric (TE) and transverse magnetic (TM) modes. Only two-dimensional electromagnetic solutions are considered here, where there is no propagation vector perpendicular to the plane of the triangle (kz = 0). The field properties just inside and outside the cavity boundary are shown to be significantly different for TE and TM field polarizations, the two cases having different dependences on the index mismatch with the exterior. For a given mode specified by particular quantum numbers, TE polarization leads to longer lifetime than TM polarization at high index mismatch, assuming that escape of evanescent boundary waves at the corners is the primary decay process.
LYAPUNOV-BASED TRANSFER BETWEEN ELLIPTIC KEPLERIAN ORBITS
We present a study of the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory specific to this problem. The construction of Lyapunov functions is based on the fact that a non-degenerate Keplerian orbit is uniquely described by its angular momentum and Laplace (- Runge-Lenz) vectors. We suggest a Lyapunov function, which gives a feedback controller such that the target elliptic orbit becomes a locally asymptotically stable periodic orbit in the closed-loop dynamics. We show how to perform a global transfer between two arbitrary elliptic orbits based on the local transfer result. Finally, a second Lyapunov function is presented that works only for circular target orbits.