| Graded-index fiber is analyzed using methods of geometric optics. The derivations of equations characterizing graded-index structures are based on the Calculus of Variations. A time integral is defined for an optical ray and is used to form Euler equations. The solution of these equations gives ray path equations and defines flight invariants. The time integral is then used to derive the equations for propagation velocity.;The new concept of power density functions in the plane of flight invariants is introduced. Power density determines the distribution of power among different propagation modes, and can be used to find the impulse response.;The optimal fiber is defined as a fiber with axial focusing points. Based on this condition, several different profiles with optimizing properties are derived, among them the family of ring fibers.;The discrete propagation modes are derived by applying the phase resonance constraint to constant phase surfaces. This provides the link between the well-known Wentzel, Kramers, Brillouin (WKB) method and geometric optics. The analysis of ray geometry also shows that the WKB method is not correct in the case of a cylindrical structure with monotonic profile.;Fiber chains are analyzed based on propagation velocity conversion at a fiber splice. It is shown that the bandwidth of a fiber chain can be broadened by choosing fibers of opposite propagation velocity characteristics. Thus, fibers with profiles different from parabolic, due to imperfections introduced in the production process, can be equalized by attaching a compensating fiber.;A new family of fibers with bi-symmetric profile functions including nearly optimal elliptical fibers is analyzed. The "tunneling" rays in a cylindrical structure are shown to escape from the fiber core if there exist a deviation from perfect cylindrical symmetry. |
