tic
close all; clc;
% For the region k_o < k_p < sqrt(u_r*e_r)*k_o
% Wavelength value lambda
lambda = 1;
% Wavenumber in free space value = k_o
k = (2*pi)/lambda;
% Permeability Value = u_r
u_r = 1;
% Permittivity Value e_r
e_r = 2.2;
% Value of kz_1
kz_1 =@(kp) sqrt((k).^2*e_r*u_r - (kp).^2);
% Value of kz_2
kz_2 = @(kp) sqrt((kp).^2 - (k).^2);
% Substrate heights = d
D = [0.02 0.04 0.06 0.08 0.10];
for i=1:numel(D)
d=D(i);
% Equations for TM(kp)
T_M = @(kp) kz_1(kp).*sin(d.*kz_1(kp)) - e_r*sqrt(kz_2(kp)).*cos(d.*kz_1(kp));
kp_root=fzero(T_M, [k , k*sqrt(e_r*u_r)])
fplot(T_M,[k k*sqrt(e_r*u_r)], 'LineWidth',3); hold on
plot(kp_root,0,'o','MarkerSize',8,'MarkerFaceColor','k');
ylim([-5 5]);
xlim([k k*sqrt(e_r*u_r)]);
grid on;
ax = gca;
ax.GridLineWidth = 2;
end; hold off
