Abstract
We study in this paper the propagation of electro-magnetic (optical) waves, guided by a medium consisting of layers of dielectric material. This problem can be reduced to an eigenvalues problem of the form Lu+ f(u)u=;u, ∈ s H1 (R){0}, where L is a linear, second order differential operator and f(u) u is a nonlinear term. We are interested in solutions (;n, d, un, d) where un,d has exactly n−1 zeroes and where\(d = \left| {u_{n,d} } \right|_{L^2 } \). (Remark that d2 is related to the power of the beam.) Under suitable conditions, such solutions exist, ∀n εN*; tin certain cases, we establish the solution for all values of d>0, while in other cases such solutions exist only when d is large enough. These results agree with what is expected from a physical point of view: either guidance occurs at all powers, or guidance occurs only at high powers, or no guidance at all occurs. Beside the existence of solutions, we analyze their behavior as d varies; tin the case where guidance occurs at all powers we show that all the families of solutions have a common bifurcation behavior.\(\lambda _{n,d} \to 0,\left| {u_{n,d} } \right|_{L^2 } \to 0\) as d→0.
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This research has been supported by the Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung.
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Ruppen, HJ. Multiple TE-modes for planar, self-focusing wave guides. Annali di Matematica pura ed applicata 172, 323–377 (1997). https://doi.org/10.1007/BF01782618
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DOI: https://doi.org/10.1007/BF01782618