First, a new method for numerical inversion of the Laplace transform is proposed. The essential point of this method is to approximate the exponential function in the Bromwich integral by the function Eec(S, a) ___a exp (a)/2 cosh (a -s). With this we can get an infinite series which gives a good approximation to the inverse-transform. In practice, we accelerate the convergence of the series by the Euler transformation. Thus in ordinary cases, only twenty to thirty terms give satisfactory results, and even a $100 pocket calculator can solve practical problems which are not so easy by the usual method. Second, the wave propagation in dispersive media is studied by using the above method. This problem has been investigated analytically by many authors. The problem is so difficult that there has been no detailed solution available other than asymptotic expressions. Some detailed numerical results by our method are given for the transient phenomena in a waveguide and the medium discussed by Sommerfeld and Brillouin. Our numerical results show that Brillouin's results should not be taken quantitatively except the first forerunner whose approximate variation is predicted by Sommerfeld correctly.
We propose an approximate method, of general applicability, for analyzing the discontinuities in slab dielectric waveguides. The method is based on replacing the unbounded configuration by a corresponding periodic multilayer structure. Hence the entire spectrum becomes discrete, and this makes the problem easier. The characteristic equation for the modal solution in a periodical multilayer dielectric waveguide is expressed in a matrix form which can be readily solved with a computer. In order to solve the discontinuity problems of slab waveguides, we expressed the reflected and transmitted fields by truncated modal expansions. The unknown amplitudes of the reflected and transmitted modes are determined by the boundary conditions. Numerical results are graphically shown including the cases which have been difficult to analyze by other methods. The accuracy of our results is checked by evaluating the relative errors and comparing with other available results.
This paper proposes new optical fibers with angularly nonuniform index distributions featured by one circular pit in the core, two circular pits in the core, and noncoaxial W‐type distribution. The field problems are analyzed by the improved point‐matching method. Waveguide parameters of single circular‐pit fibers, double circular‐pit fibers, and noncoaxial W‐type fibers that provide the largest modal birefringence are investigated. It is found that (1) the maximum modal birefringence Bmax for single circular‐pit fibers and noncoaxial W‐type fibers is much larger than those for double circular pit fibers, (2) noncoaxial W‐type fibers exhibit larger modal birefringence than single circular‐pit fibers in the range Δn < 1.8% (Δn is relative index difference), and (3) the value Bmax of 1.7 × 10−4 has been attained for Δn = 0.42%. These results are quite reliable because (1) the relative error of the modal birefringence computed by our method is less than 10−4, and (2) this new method never produces ghost solutions.
The electromagnetic waves propagating along an open waveguide can be classified in several ways. For instance: (1) guided modes and leaky modes; (2) real modes and complex modes; and (3) proper modes and improper modes. However, their mutual relationships are not clear. In addition, the physical meaning of the leaky mode is not well understood in comparison with the guided mode. In this paper, the relationship of the modes propagating in a dielectric slab waveguide is investigated. First, from the dispersion equation, the guided mode cutoff frequency and the leaky mode cutoff frequency are obtained so that the regions of the guided mode and the leaky mode are found. It is shown also that there exists an improper mode region with real propagation constants between the guide and leaky mode regions. This improper mode does not decay in the direction of propagation and the amplitude increases away from the waveguide along a straight line with an arbitrary angle from the propagation direction. In this region, the group velocity may exceed the speed of light in free space or become infinite. Thus, in this paper, this region is called the anomalous group velocity region. The width of this region is inversely proportional to the mode number and the waveguide width.
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