Lecture 29: 17 November 2010 In a waveguide, or any system that is invariant along one dimension (say y), we can always find separable eigenfunctions of Âu=∂2u/∂t2. That is, we look for solutions of the form uk(x,z) ei(ky-ωt),which are eigenfunctions of  with eigenvalue -ω2. These are solutions to the full problem, with each value of k giving us different solutions uk and ω(k). We can then build any arbitrary solution u via a superposition of these (much like a Fourier transform, writing any y dependence as a sum of eiky sinusoids). For each k, the function uk(x,z) (which does not depend on y) satisfies Âeikyuk=-ω(k)2eikyuk, from which we derive the reduced eigenproblem Âkuk=-ω(k)2uk, where Âk=e-ikyÂeiky is an operator with no y derivatives and no y dependence: we have reduce the problem to one fewer spatial dimension. Showed that Âk is self-adjoint and definite if  is. , and showed (by the usual product rule for is (ikyMore concretely, started with the operator differentiation) that the corresponding reduced operator e-ikyxz derivatives and k is the vector in the y direction with magnitude k. Hence, if we take xz+ik) where xz has onlyeÂ=b ], the corresponding reduced operator is Âk=b(·[axz+ik)·[a(xz+ik)], whic simplifies to Âk=Â-abk2 (since the cross terms have zero dot products). This is even more obviously self-adjoint and negative-definite if  is, and in fact is negative definite for k≠0 even if  is only negative semidefinite (e.g. for Neumann). If we apply the min-max principle to Âk=Â-abk2, we find as usual that the lowest-ω(k) mode "wants" to not oscillate too much and to be concentrated in the regions of smaller a and b. However, the balance between these two considerations depends on k: as k2 increases, we see that the tendency to concentrate in the small-ab regions (small-c regions) becomes stronger. Numerical example (see waveguide.m) of modes concentrating in a region of smaller c2=ab in two dimensions invariant in y, corresponding to a 1d reduced eigenproblem. This is what we referred to before as a "total internal reflection" waveguide. Plotted the dispersion relation ω(k) and the cone of solutions in the surrounding larger-c region. As predicted, we obtain guided modes below the cone, and as k increases both the number of guided modes and their localization increases. More on this example in the next lecture. Further reading: Chapter 3 of our book on electromagnetism covers the reduced eigenproblem in the context of Maxwell's equations.MIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit:
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