Computer EngineeringPaper details:Course : Optical Computer SystemELG5103 Optical Communications Systems Assignment 2 Waves in a Slab WaveguideHint: The electric field distribution varies depending on the confinement factorand the dispersion relation. To find the electric field distribution inside the coreand the cladding you will need to find K & y which will involve solving thedispersion relation for a particular value of V – this may be done with a few linesof Matlab. See supplementary material.3) To keep the notation light-weight it may be assumed that one is only concerned withTE-like modes with transverse amplitude profiles adequately described by scalarvalued eigenfunctions (pp associated to an eigenvalue 3,, corresponding to thepropagation constant of the mode i.e.up (x, z) = (pp (x) exp(i,6’pz)The (pp are orthonormal with respect to an inner product:(‘pp”pq) = 6mThe inner product may take different forms but it is always linear in the first argumentand conjugate linear in the second argument and for purpose of this problem may betaken1 as:(em wq) – I epwqu – 6pc:The set of eigenfunctions is complete.Suppose light is launched into a waveguide corresponding a total forward going field1? , exploit the orthonormality of the modes relative to the inner product to find thecoefficients of the expansion of the field ‘1’ as linear combination of the modes. WhatI The electric 5,, and magnetic fields of propagating modes of a dielectric waveguide satisfy the orthogonality condition:fi(£,,xfl;+5; xup).ds= (SWNo longitudinal component takes part in this integral and so one may considerthe transverse components only.In the special case of TE & TM modes, the TE mode is always orthogonal to a TM mode and vice versa. It is therefore only necessaryto consider the orthogonality within the set of TE and TM modes:For TE one notes that only the 3 component of H x contributes and:as),HI. O( E = IKE),For TM one notes that only the 3 component of E , contributes and:1 as 15 -” = ‘ -Ex cc 722(x) dz ll? 122(x) yHence with suitable normalization one finds:(‘PpKM : [(pp
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