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www.ece.rutgers.edu/∼orfanidi/ewa 285 286 Electromagnetic Waves & Antennas – S. J. Orfanidis – June 21, 2004 v = u tan(u − R m ) v 2 + u 2 = R 2 (mth mode) (8.11.29) If one had an approximate solution u, v for the mth mode, one could refine it by using Newton’s method, which converges very fast provided it is close to the true solution. Just such an approximate solution, accurate to within one percent of the true solution, was given by Lotspeich [464]. Without going into the detailed justification of this method, the approximation is as follows: u = R m + w 1 (m)u 1 (m)+w 2 (m)u 2 (m) , m = 0, 1,...,M (8.11.30) where u 1 (m), u 2 (m) are approximate solutions near and far from the cutoff R m , and w 1 (m), w 2 (m) are weighting factors: Fig. 8.11.2 Even and odd TE modes at different frequencies. Given a value of R, we determine M as that integer satisfying Eq. (8.11.24), or, M ≤ 2R/π
www.ece.rutgers.edu/∼orfanidi/ewa 287 288 Electromagnetic Waves & Antennas – S. J. Orfanidis – June 21, 2004 [be,kc,ac,fc,err] = dguide(f,a,n1,n2,Nit); % dielectric slab guide where f is in GHz, a in cm, and β, k c ,α c in cm −1 . The quantity f c is the vector of the M + 1 cutoff frequencies defined by the branch edges R m = mπ/2, that is, R m = ω m aN A /c 0 = 2πf m aN A /c 0 = mπ/2, or, f m = mc 0 , m = 0, 1,...,M (8.11.35) 4aN A The meaning of f m is that there are m + 1 propagating modes for each f is in the interval f m ≤ f
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