exercise 3.13

mu0=4*pi*10^-7; e0= 8.8539e-12;

f0=20e9                                                     % Hz

c0=299792486;                                      % m/s through vacuum or air

a=.4e-2;                                                   % radius [m]

mur=1;

er=1.5;                                                    % relative permittivity

etha0=377;etha=etha0/er^.5;

lambda0=c0/f0                                        % wave length in free space

tand=.0002;

sigma_Cu=5.813e7;  % S/m

% sigma_Au=4.1e7;  % S/m

% sigma_Ag=6.173e7;  % S/m

                

N=5;n=[1:1:N];                                        % 5 1st modes

k_TE=2*pi*f0*(mu0*e0*mur*er)^.5       % wave number

k_TM=2*pi*f0*(mu0*e0*mur*er)^.5      % k_TE k_TM same as k0

k0=2*pi/lambda0                                        % wave number

 

x=[0:.001:15];                          % finding J0(x) and J0(x)' zeros: p and p1

M=5;N=5;y=zeros(M(end),numel(x));

for k=0:1:M-1

    y(k+1,:)=besselj(k,x);

end

p={};                                          % p: J zeros tor TM cut-offs

for k=1:1:M

    [nx0,ny0]=intersections(x,y(k,:),x,zeros(1,numel(x)));

    p{k}=nx0;

end

p{:}

for k=1:1:numel(p)                % removing origin roots of p

    Lp=p{k}                              % to avoid p(i,j) TMij indices mismatch

    Lp(Lp==0)=[]

    p{k}=Lp

end

 

yd=diff(y')';x(1)=[];            % 1st deriv of 5 lowest Bessel 1st kind

p1={};                                    % p1: J' zeros for TE cut-offs

for k=1:1:M

    [nx0d,ny0d]=intersections(x,yd(k,:),x,zeros(1,numel(x)));

% TE find zeros 1st derivative

    p1{k}=nx0d;

end

p1{:};

for k=1:1:numel(p1)             % removing origin roots of p1

    Lp1=p1{k}                            % to avoid p1(i,j) TEij indices mismatch

    Lp1(Lp1==0)=[]

    p1{k}=Lp1

end

 

L4=[0 0 0];                                % TE modes and fc in L5: n m p'_nm

for k=1:1:M                               % n: order Bessel function

    L05=p1{:,k}';                        % m: root numeral

    for s=1:1:length(L05)            % p'_nm dJ(x)/dx roots

        L4=[L4;[(k-1) s L05(s)]];

    end

end

L4(1,:)=[];

L4=[L4(:,1) L4(:,2) L4(:,3)*c0/er^.5*1/(2*pi*a)];

L4_3=L4(:,3);

[fc_TE_sorted,nfc_TE_sorted]=sort(L4_3);

L5=L4(nfc_TE_sorted,:);             % L5 contains sorted fc_TE with TE indices

uint64(L5)                                      % fc TE sorted including null TE cut-offs

fc_TE=L5(:,3);                               % remove null TE cut-offs

[nfcTEz,v]=find(fc_TE==0);

L5(nfcTEz,:)=[];fc_TE(nfcTEz)=[];

fc_TE_table_names={'n' 'm' 'fc' 'mode_tag'}                     

% generating table with TE cut-off frequencies, indices, and TE tag

tag_TE=repmat('TE',size(L5,1),1)

fc_TE_table=table(uint64(L5(:,1)),uint64(L5(:,2)),...

L5(:,3),tag_TE,'VariableNames',fc_TE_table_names)

 

L6=[0 0 0];                                    % TM modes and fc in L7: m n p_nm

for k=1:1:M                                   % n: order Bessel function

    L07=p{:,k}';                             % m: root numeral

    for s=1:1:length(L07)                 % p'_nm dJ(x)/dx roots

        L6=[L6;[(k-1) s L07(s)]]; 

    end

end

L6(1,:)=[];

L6=[L6(:,1) L6(:,2) L6(:,3)*c0/(er^.5*2*pi*a)];

L6_3=L6(:,3);

[fc_TM_sorted,nfc_TM_sorted]=sort(L6_3);

L7=L6(nfc_TM_sorted,:);                                      

% L7 contains sorted fc_TM with TM indices

uint64(L7)                                      % fc TM sorted included null TM cut-offs

fc_TM=L7(:,3);                              % remove null TM cut-offs

[nfcTMz,v]=find(fc_TM==0);

L7(nfcTMz,:)=[];fc_TM(nfcTMz)=[];

fc_TM_table_names={'n' 'm' 'fc' 'mode_tag'}                    

% generating table with TE cut-off frequencies, indices, and TE tag

tag_TM=repmat('TM',size(L7,1),1)

fc_TM_table=table(uint64(L7(:,1)),uint64(L7(:,2)),L7(:,3),tag_TM,...

                                                         'VariableNAmes',fc_TM_table_names)

 

fc_TE_TM_table_names={'n' 'm' 'fc' 'mode_tag'}                    

% generating table with TE cut-off frequencies, indices, and TE tag

tag_=[repmat('TE',size(L5,1),1); repmat('TM',size(L7,1),1)]

fc_TE_TM_t=table([uint64(L5(:,1));uint64(L7(:,1))],...

                           [uint64(L5(:,2));uint64(L7(:,2))],...

                           [L5(:,3);L7(:,3)],...

                           tag_,'VariableNames',fc_TE_TM_table_names)

fc_TE_TM_table=sortrows(fc_TE_TM_t,'fc')

numerals_table=table([1:1:numel(L5(:,1))+numel(L7(:,1))]',…

                                                            'VariableNames',{'pos'})

fc_TE_TM_table=[numerals_table fc_TE_TM_table]

k_sort_fc=1;

while f0>fc_TE_TM_table.fc(k_sort_fc) 

        k_sort_fc=k_sort_fc+1;

        if k_sort_fc>numel(fc_TE_TM_table.fc)

            break

        end

end

fc_TE_TM_table