# Article

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Keywords:
Schrödinger's equation; waveguides
Summary:
We discuss the propagation of electromagnetic waves of a special form through an inhomogeneous isotropic medium which has a cylindrical symmetry and a nonlinear dielectric response. For the case where this response is of self-focusing type the problem is treated in [1]. Here we continue this study by dealing with a defocusing dielectric response. This tends to inhibit the guidance properties of the medium and so guidance can only be expected provided that the cylindrical stratification is such that guidance would occur for the linear response that is obtained in the limit of zero field strength. The guided modes that we seek correspond to solutions of the boundary value problem $-u'' + \frac 34 \frac u{r^2} - q(r) u + p( r, u ) u = \lambda u$ for $r > 0$ with $u \in H^1_0 ( 0, \infty )$ and its linearisation is $-u'' + \frac 34 \frac u{r^2} - q( r ) u = \lambda u$ with $u \in H_0^1 ( 0, \infty )$. This linear problem has the interval $[0, \infty )$ as its essential spectrum and the requirement that guidance should occur in the limit of zero field strength leads us to suppose that it has at least one negative eigenvalue. Solutions of the nonlinear problem are then obtained by bifurcation from such an eigenvalue. The main interest concerns the global behaviour of a branch of solutions since this determines the principal features of the waveguide. If the branch is bounded in $L^2 ( 0, \infty )$ there is an upper limit to the intensity of the guided beams (high-power cut-off), whereas if the branch is unbounded in $L^2 ( 0, \infty )$ then guidance is possible at arbitrarily high intensities. Our results show how these behaviours depend upon the properties of dielectric response.
References:
[1] Stuart C.A.: Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. Arch. Rat. Mech. Anal. 113 (1991), 65-96. MR 1079182 | Zbl 0745.35044
[2] Stuart C.A.: Global properties of components of solutions of nonlinear second order ordinary differential equations on the half-line. Ann. Sc. Norm. Sup. Pisa II (1975), 265-286. MR 0380013
[3] Stuart C.A.: The behaviour of branches of solutions of nonlinear eigenvalue problems. Rend. Ist. Matem. Univ. Trieste XIX (1987), 139-154. MR 0988378 | Zbl 0667.34027
[4] Kato T.: Perturbation Theory for Linear Operators. Springer-Verlag Berlin (1966). MR 0203473 | Zbl 0148.12601
[5] Weidmann J.: Linear Operators in Hilbert Space. Springer-Verlag Berlin (1980). MR 0566954
[6] Eastham, M.S.P.: Theory of Ordinary Differential Equations. Van Nostrand (1970). Zbl 0195.37001
[7] Akhmanov R.V., Khokhlov R.V., Sukhorukov A.P.: Self-focusing, self-defocusing and self-modulation of laser beams. Laser Handbook (ed. by F.T. Arecchi and E.O. Schulz Dubois), North Holland, Amsterdam, 1972.
[8] Mathew J.G.H., Kar A.K., Heckenberg N.R., Galbraigth I.: Time resolved self-defocusing in InSb at room temperature. IEEE J. Quantum Elect. 21 (1985), 94-99.
[9] Stegeman G.I., Wright E.M., Seaton C.T., Moloney J.V., Shen T.-P., Maradudin A.A., Wallis R.F.: Nonlinear slab-guided waves in non-Kerr-like media. IEEE J. Quantum Elect. 22 (1986), 977-983.
[10] Stuart C.A.: Guidance Properties of Nonlinear Planar Waveguides. Arch. Rational Mech. Anal. 125 (1993), 145-200. MR 1245069 | Zbl 0801.35136

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