Svetislav Savović

Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions

Abstract A finite difference method is used to solve the one-dimensional Stefan problem with periodic Dirichlet boundary condition. The temperature distribution, the position of the moving boundary and its velocity are evaluated. It is shown that, for given oscillation frequency, both the size of the domain and the oscillation amplitude of the periodically oscillating surface temperature, strongly influence the temperature distribution and the boundary movement. Furthermore, good agreement between the present finite difference results and numerical results obtained previously using the nodal integral method is seen.

Investigation of mode coupling in step index plastic optical fibers using the power flow equation

Using the power flow equation, we have examined the mode coupling caused by intrinsic perturbation effects of the step index plastic optical fiber. A numerical solution has been obtained by the explicit finite difference method. Results show the state of mode coupling along the fiber. They indicate that the equilibrium mode distribution is achieved at much shorter lengths compared to the case with glass fibers.

Numerical solution of the power flow equation in step-index plastic optical fibers

The numerical solution of the complete power flow equation is reported and employed to investigate the state of mode coupling along a step-index plastic optical fiber. This solution is based on the explicit finite-difference method and, in contrast to earlier solutions, does not neglect absorption and scattering loss. It is the only solution that can accommodate any input condition throughout the entire range of feasible input angles without the need for restriction to those angles that are sufficiently far away from critical. Our results for the field patterns at different locations along one type of fiber are in agreement with reported measurements earlier. Furthermore, the length of fiber required for achieving a steady-state mode distribution matches the analytical solution that is available for such distribution as a special case. Mode coupling in plastic fibers is known to affect fiber-optic power delivery, data transmission, and sensing systems.

Influence of numerical aperture on mode coupling in step-index plastic optical fibers

Using the power-flow equation, we have examined the state of mode coupling in step-index plastic optical fibers with different numerical apertures. Our results confirm that the coupling rates vary with the coupling coefficient of the fibers as the dominant parameter, especially in the early stage of coupling near the input fiber end. However, we show that the fibers numerical aperture has a significant influence on later stages of this process. Consequently, equilibrium mode distribution and steady-state distribution are achieved at overall fiber lengths that depend on both of these factors. As one of our examples demonstrates, it is possible for the coupling length of a high-aperture fiber to be similar to that of a low-aperture fiber despite the three-times-larger coupling coefficient of the former.

Optical power flow in plastic-clad silica fibers

Using the time-independent power-flow equation, we have examined the mode coupling caused by intrinsic perturbation effects of step-index plastic clad silica fiber carrying more than 10(5) modes. Result show that the equilibrium mode distribution for this fiber is achieved at a length of approximate 550 m, which is longer than reported previously. While this coupling length is much longer than that of plastic optical fibers, it is sorter than that of all-glass fibers.

Method for calculating the coupling coefficient in step-index optical fibers

A simple method is proposed for determining the mode coupling coefficient D in step-index multimode optical fibers. It only requires observation of the far-field output pattern for one fiber length with the input light launched centrally along the fiber axis (theta(0)=0). For illustration, the coupling coefficient determined by this simple method for a step-index plastic optical fiber was used to calculate the coupling length L(c) at which the equilibrium mode distribution is achieved, and length z(s) at which the steady-state distribution is achieved. Our results are in good agreement with experimental results reported earlier.

Mode coupling in strained and unstrained step-index glass optical fibers

By using the power flow equation, we have examined the state of mode coupling in strained and unstrained step-index glass optical fibers. Strained fibers show stronger mode coupling than their unstrained counterparts of the same type. As a result, the coupling length where equilibrium mode distribution is achieved and the length of fiber required for achieving the steady-state mode distribution are shorter for strained than for unstrained fibers.

Nodal Integral and Finite Difference Solution of One-Dimensional Stefan Problem

The nodal integral and finite difference methods are useful in the solution of one-dimensional Stefan problems describing the melting process. However, very few explicit analytical solutions are available in the literature for such problems, particularly with time-dependent boundary conditions. Benchmark cases are presented involving two test examples with the aim of producing very high accuracy when validated against the exact solutions. Test example 1 (time-independent boundary conditions) is followed by the more difficult test example 2 (time-dependent boundary conditions). The temperature distribution, position of the moving boundary and the velocity are evaluated

Solution of mode coupling in step-index optical fibers by the Fokker-Planck equation and the Langevin equation

The power-flow equation is approximated by the Fokker-Planck equation that is further transformed into a stochastic differential (Langevin) equation, resulting in an efficient method for the estimation of the state of mode coupling along step-index optical fibers caused by their intrinsic perturbation effects. The inherently stochastic nature of these effects is thus fully recognized mathematically. The numerical integration is based on the computer-simulated Langevin force. The solution matches the solution of the power-flow equation reported previously. Conceptually important steps of this work include (i) the expression of the power-flow equation in a form of the diffusion equation that is known to represent the solution of the stochastic differential equation describing processes with random perturbations and (ii) the recognition that mode coupling in multimode optical fibers is caused by random perturbations.

Equilibrium mode distribution and steady-state distribution in 100–400 μm core step-index silica optical fibers

Using the power flow equation, the state of mode coupling in 100-400 μm core step-index silica optical fibers is investigated in this article. Results show the coupling length L(c) at which the equilibrium mode distribution is achieved and the length z(s) of the fiber required for achieving the steady-state mode distribution. Functional dependences of these lengths on the core radius and wavelength are also given. Results agree well with those obtained using a long-established calculation method. Since large core silica optical fibers are used at short distances (usually at lengths of up to 10 m), the light they transmit is at the stage of coupling that is far from the equilibrium and steady-state mode distributions.