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T_matrix_for_Multiple_Cylinders.m
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T_matrix_for_Multiple_Cylinders.m
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% This Matlab code calculates the scattering field of multiple multilayer 2-D plasma cylinders using T-matrix method. This method can also be used
% for approximately estimating the scattering field of long and thin 3-D cylinders.
% The T-matrix of a single cylinder can be calculated by generalized reflection coefficient method, then a T-matrix equation is formed for the multiple scattering problem.
% Notice: Large number of Mie series is needed for larger-scale problem. T-matrix method is suitable for calculating the scattering of many compact (~10 wavelength) scatters.
% Reference: S. S. A. Yuan, Z. H. Lin, L. -B. Lv, S. -J. Hao and W. E. I. Sha, "Investigating the Scattering Characteristics of Artificial Field-Aligned Irregularities Based on T-Matrix Algorithm," in IEEE Journal on Multiscale and Multiphysics Computational Techniques, vol. 8, pp. 147-157, 2023.
% Shuai S. A. Yuan
% Email: shuaiyuan1997@zju.edu.cn
function T_matrix_for_Multiple_Cylinders
clc;clear
%% input
Wavelength=55; % Wavelength (m)
Polarization=1; % Polarizations (1-TM wave, 2-TE wave, 3-left circularly polarized wave, 4-right circularly polarized wave)
Radius=55;% Radius of a single cylinder, the cylinders are along the z axis
Layers=10;% Number of layers of a cylinder, the most outside layer is air layer
electron_density=6.5e11;
central_density=5*electron_density; % central electron density
Angle=90; % Angle between the incident plane wave and z axis
% Number and positons of the cylinders
%X_position=[0,0,0,3,-3,-3,3,3,-3]*Radius;
%Y_position=[0,3,-3,0,0,-3,3,-3,3]*Radius;
%X_position=[0]*Radius; % Single cylinder x
%Y_position=[0]*Radius; % Single cylinder y
X_position=[-3,3]*Radius;
Y_position=[-3,3]*Radius;
%% Parameters
ind_x=Wavelength;
Pn=length(X_position);
%incident angle
incident_angle=Angle/180*pi;
phi_angle=0/180*pi;
% basic parameter
epsc=1/(4*pi*9*10.^9); % permittivity in free space
murc=4*pi*10.^(-7); % permeability in free space
electron_charge=1.60217*10.^(-19);% electron charge (coulombs)
electron_mass=9.10938*10.^(-31); %electron mass (kg)
collision_frequency=1e3; % loss in drude model (rad/s)
num_layer=Layers; %number of layers
FAI_radius=Radius; %radius of FAI (m)
FAI_r=linspace(eps,eps+FAI_radius,num_layer);
FAI_w=FAI_radius*sqrt(-1/(log(electron_density/central_density))); % gaussian distribution coefficients
layer_density=central_density*exp(-FAI_r.^2/FAI_w^2);
% relative permittivity
Air=ones(1,length(ind_x))*1; % air
%Si_epr=3.4^2*ones(1,length(x))*1; % silicon
%% Wavenumber of incident wave
theta=incident_angle;
phi=phi_angle;
wavelength=ind_x;%m
k_0=2*pi/wavelength;
k_z=k_0*cos(theta);
k_rho=k_0*sin(theta);
k_x=k_rho*cos(phi);
k_y=k_rho*sin(phi);
omega=2*pi*3*10^8./wavelength;%
MAX=ceil(2*pi*FAI_radius/wavelength+4.05*(2*pi*FAI_radius/wavelength)^(1/3)+2); % Number of Mie series
L=num_layer;
% coefficient a
Z0=sqrt(murc/epsc);
% outmost layer coefficient for cylinder
% a at L layer, a 1x2 matrix amplitude for TE and TM
a=zeros(2,L,MAX);
for n=-MAX:1:MAX % for all harmonics % - infinite to + infinite, different from spherical case
%a(:,L,n+MAX+1)=[1j^(-n);1j^(-n)/Z0]; %
a(:,L,n+MAX+1)=[0;(1j^(-n)/Z0)]; % TM (1j^(n)) and TE (1j^(n)/Z0)
end%
%% ¹ãÒå·´ÉäϵÊý
for fre=1:length(wavelength)
fre
%eps_array=[Si_epr(fre),Air(fre)]; % relative permittivity (inner to outer)
%eps_array=[eps_save(fre),Air(fre),eps_save(fre),Air(fre)]; % relative permittivity (inner to outer)
%eps_array=[eps_save(fre),Air(fre),eps_save(fre),Air(fre)]; % relative permittivity (inner to outer)
mu_array=ones(1,num_layer); % relative permeability
rho_array=FAI_r(2:end); % radius (inner to outer)
% drude model for ionosphere plasma
plasma_fre=sqrt(electron_charge.^2*layer_density/epsc/electron_mass);
eps_plasma=1-(electron_charge.^2*layer_density/epsc/electron_mass/(omega(fre).^2+1i*omega(fre)*collision_frequency));%not relative
%eps_plasma=ones(1,num_layer);
%eps_plasma(1:L-1)=3;
eps_plasma(L)=1; %use air for most out layer
k_array=sqrt(mu_array.*eps_plasma)*2*pi/(wavelength(fre));
kz_array=k_array*cos(incident_angle);
krho_array=k_array*sin(incident_angle);
eps_array=eps_plasma*epsc;
%eps_array=eps_array*epsc;
mu_array=mu_array*murc;
rho_array=rho_array;%m
% inverse the order
inv_k_arr=k_array(end:-1:1);
inv_kz_arr=kz_array(end:-1:1);
inv_krho_arr=krho_array(end:-1:1);
inv_eps_arr=eps_array(end:-1:1);
inv_mu_arr=mu_array(end:-1:1);
inv_rho_arr=rho_array(end:-1:1);
% Outgoing wave R and T coef matrix (inner to outer)
for m=1:L-1;
for n=-MAX:1:MAX
R_Out(:,:,m,n+MAX+1)=... %inv(...) = D matrix
inv(J_matrix(n,krho_array(m),kz_array(m),rho_array(m),eps_array(m),omega(fre),mu_array(m))...
*cb2(n,krho_array(m+1),rho_array(m))-H_matrix(n,krho_array(m+1),kz_array(m+1),rho_array(m),...
eps_array(m+1),omega(fre),mu_array(m+1))*cb1(n,krho_array(m),rho_array(m)))*( cb2(n,...
krho_array(m),rho_array(m))*H_matrix(n,krho_array(m+1),kz_array(m+1),rho_array(m),...
eps_array(m+1),omega(fre),mu_array(m+1))- cb2(n,krho_array(m+1),rho_array(m))*H_matrix(n,...
krho_array(m),kz_array(m),rho_array(m),eps_array(m),omega(fre),mu_array(m)) );
T_Out(:,:,m,n+MAX+1)=...
inv(J_matrix(n,krho_array(m),kz_array(m),rho_array(m),eps_array(m),omega(fre),mu_array(m))...
*cb2(n,krho_array(m+1),rho_array(m))-H_matrix(n,krho_array(m+1),kz_array(m+1),rho_array(m),...
eps_array(m+1),omega(fre),mu_array(m+1))*cb1(n,krho_array(m),rho_array(m)))*( cb2(n,...
krho_array(m),rho_array(m))*J_matrix(n,krho_array(m),kz_array(m),rho_array(m),...
eps_array(m),omega(fre),mu_array(m))- cb1(n,krho_array(m),rho_array(m))*H_matrix(n,...
krho_array(m),kz_array(m),rho_array(m),eps_array(m),omega(fre),mu_array(m)) );
T_Out_test(:,:,m,n+MAX+1)=...
inv(J_matrix(n,krho_array(m),kz_array(m),rho_array(m),eps_array(m),omega(fre),mu_array(m))...
*cb2(n,krho_array(m+1),rho_array(m))-H_matrix(n,krho_array(m+1),kz_array(m+1),rho_array(m),...
eps_array(m+1),omega(fre),mu_array(m+1))*cb1(n,krho_array(m),rho_array(m)))*[eps_array(m),0;...
0,-mu_array(m)]*2*omega(fre)/pi/krho_array(m)^2/rho_array(m);
end
end
% Standing wave R and T coef matrix (outer to inner)
% R_43 R_32 R_21
for m=1:L-1;
for n=-MAX:1:MAX
R_In_inv(:,:,m,n+MAX+1)=... %inverse the eps mu k
inv(J_matrix(n,inv_krho_arr(m+1),inv_kz_arr(m+1),inv_rho_arr(m),inv_eps_arr(m+1),omega(fre),inv_mu_arr(m+1))...
*cb2(n,inv_krho_arr(m),inv_rho_arr(m))-H_matrix(n,inv_krho_arr(m),inv_kz_arr(m),inv_rho_arr(m),...
inv_eps_arr(m),omega(fre),inv_mu_arr(m))*cb1(n,inv_krho_arr(m+1),inv_rho_arr(m)))*( cb1(n,...
inv_krho_arr(m+1),inv_rho_arr(m))*J_matrix(n,inv_krho_arr(m),inv_kz_arr(m),inv_rho_arr(m),...
inv_eps_arr(m),omega(fre),inv_mu_arr(m))- cb1(n,inv_krho_arr(m),inv_rho_arr(m))*J_matrix(n,...
inv_krho_arr(m+1),inv_kz_arr(m+1),inv_rho_arr(m),inv_eps_arr(m+1),omega(fre),inv_mu_arr(m+1)) );
T_In_inv(:,:,m,n+MAX+1)=...
inv(J_matrix(n,inv_krho_arr(m+1),inv_kz_arr(m+1),inv_rho_arr(m),inv_eps_arr(m+1),omega(fre),inv_mu_arr(m+1))...
*cb2(n,inv_krho_arr(m),inv_rho_arr(m))-H_matrix(n,inv_krho_arr(m),inv_kz_arr(m),inv_rho_arr(m),...
inv_eps_arr(m),omega(fre),inv_mu_arr(m))*cb1(n,inv_krho_arr(m+1),inv_rho_arr(m)))*[inv_eps_arr(m),0;...
0,-inv_mu_arr(m)]*2*omega(fre)/pi/inv_krho_arr(m)^2/inv_rho_arr(m);
%notice the region and inverse, layer order exchange
end
end
% Standing wave R and T coef matrix£¨inner to outer£©
% R_21 R_32 R_43
R_In=R_In_inv(:,:,end:-1:1,:);
T_In=T_In_inv(:,:,end:-1:1,:);
R_In_test=R_In(:,:,end,6);
% Generalized R and T coef matrix Ref
Gen_R(:,:,1,:)=R_In(:,:,1,:);%first layer
for m=2:L-1
for n=-MAX:1:MAX
Gen_R(:,:,m,n+MAX+1)=R_In(:,:,m,n+MAX+1)+(T_Out(:,:,m,n+MAX+1)*Gen_R(:,:,m-1,n+MAX+1)...
*inv(eye(2)-R_Out(:,:,m,n+MAX+1)*Gen_R(:,:,m-1,n+MAX+1))*T_In(:,:,m,n+MAX+1));
end
end
%test_Tout(:,:,:)=T_Out(:,:,:,1);
test_Gen_R(:,:)=Gen_R(:,:,end,10);
% get coefficients ampititude
for m=L-1:-1:2
for n=-MAX:1:MAX
a(:,m,n+MAX+1)=inv(eye(2)-R_Out(:,:,m,n+MAX+1)*Gen_R(:,:,m-1,n+MAX+1))*T_In(:,:,m,n+MAX+1)*a(:,m+1,n+MAX+1);
end
end
for n=-MAX:1:MAX %most inner interface
a(:,1,n+MAX+1)=T_In(:,:,1,n+MAX+1)*a(:,2,n+MAX+1);
end
test_a(:,:,:)=a(:,1,1);
%% b vector
b=zeros((2*MAX+1)*2,1); % Including TM and TE
b_vector=zeros((2*MAX+1)*2*Pn,1);% Final b vector
for n=-MAX:1:MAX % for all harmonics
pol_index=[1,0,-1j,1j;0,1,1,1];
b(n+MAX+1)=1j^(-n)*pol_index(1,Polarization);%1j^(-n)
b(n+3*MAX+2)=1j^(-n)/Z0*pol_index(2,Polarization);%1j^(-n)/Z0
% b_test(n+MAX+1)=a(1,L,n+MAX+1);
% a_test(n+MAX+1)=Gen_R(1,1,L-1,n+MAX+1)*a(1,L,n+MAX+1);
end
for p=1:Pn
b_vector((p-1)*(4*MAX+2)+1:p*(4*MAX+2))=b*exp(-1j*(k_x*X_position(p)+k_y*Y_position(p)));
end
%% T matrix
T_i=zeros((2*MAX+1)*2,(2*MAX+1)*2);
T_matrix=zeros((2*MAX+1)*2*Pn,(2*MAX+1)*2*Pn);
for n=-MAX:1:MAX % for all harmonics
T_MM(n+MAX+1)=Gen_R(1,1,L-1,n+MAX+1);
T_MN(n+MAX+1)=Gen_R(1,2,L-1,n+MAX+1);
T_NM(n+MAX+1)=Gen_R(2,1,L-1,n+MAX+1);
T_NN(n+MAX+1)=Gen_R(2,2,L-1,n+MAX+1);
end
T_i=[diag(T_MM),diag(T_MN);diag(T_NM),diag(T_NN)];% T matrix of each scatterer
for pp=1:Pn % Phase difference between each scatterer
for qq=1:Pn
if pp==qq
T_matrix((pp-1)*(4*MAX+2)+1:pp*(4*MAX+2),(qq-1)*(4*MAX+2)+1:qq*(4*MAX+2))=T_i;
end
end
end
%% Translation matrix
Alpha_matrix=zeros((2*MAX+1)*2*Pn,(2*MAX+1)*2*Pn);
for pp=1:Pn
for qq=1:Pn
if pp~=qq
Alpha_matrix((pp-1)*(4*MAX+2)+1:pp*(4*MAX+2),(qq-1)*(4*MAX+2)+1:qq*(4*MAX+2))=...
translation_matrix(X_position(pp),Y_position(pp),X_position(qq),Y_position(qq),MAX,k_rho);
T_test=translation_matrix(X_position(pp),Y_position(pp),X_position(qq),Y_position(qq),MAX,k_rho);
end
end
end
%% Matrix inversion
matrix_test=T_matrix*Alpha_matrix;
% rank_test=rank(matrix_test);
a_vector=(eye((2*MAX+1)*2*Pn)-T_matrix*Alpha_matrix)\(T_matrix*b_vector);
a_vector_t=a_vector(1:(4*MAX+2));
a_vector_sum=zeros(4*MAX+2,1);
for tt=1:Pn
a_vector_sum=a_vector_sum+translation_matrix2(0,0,X_position(tt),Y_position(tt),MAX,k_rho)*...
a_vector((tt-1)*(4*MAX+2)+1:tt*(4*MAX+2));
end%
%% RCS calculation
% get near field (sample)
rrho=linspace(eps,2*rho_array(end)+eps,300);
%zz=linspace(-2*rho_array(end)+eps,2*rho_array(end)+eps,round(4*rho_array(end)/0.2*10^9));
zz=[0];
phi=linspace(eps,2*pi+eps,360);
%phi=[0,pi];
% e-field components
Field_z=zeros(2,length(rrho),length(zz),length(phi));
Farfield_z=zeros(2,length(phi));
Field_phi=zeros(2,length(rrho),length(zz),length(phi));
Field_rho=zeros(2,length(rrho),length(zz),length(phi));
% calculate Ez and Hz of each layer
% most outer part % air
s=L;
R_test=Gen_R(:,:,s-1,1);
% farfield E_z for RCS calculation
s=L;
for q=-MAX:1:MAX % order
for t=1:length(phi)% phi %
% Farfield_z(:,t)=Farfield_z(:,t)+2/(sqrt(krho_array(s)))*exp(1j*q*phi(t)-1j*q/2*pi)...
% *Gen_R(:,:,s-1,q+MAX+1)*a(:,s,q+MAX+1);
Farfield_z(:,t)=Farfield_z(:,t)+exp(1j*q*phi(t)+1j*q/2*pi)...
*Gen_R(:,:,s-1,q+MAX+1)*a(:,s,q+MAX+1);
exp_test(q+MAX+1)=exp(1j*q*phi(t)-1j*q/2*pi);
% the tangential field is very small when rho -> infinite
end
end
end
%% RCS£¬ T matrix method
RCS_TM=zeros(1,(2*MAX+1));
RCS_TE=zeros(1,(2*MAX+1));
for t=1:length(phi) % phi %
for q=-MAX:1:MAX %
if abs(q)<=MAX
RCS_TM(q+MAX+1)=exp(1j*q*phi(t)+1j*q/2*pi);
RCS_TE(q+MAX+1)=exp(1j*q*phi(t)+1j*q/2*pi)*Z0;
end
end
RCS(t)=2/k_rho/sin(theta)*(abs(RCS_TM*a_vector_sum(1:2*MAX+1)).^2+...
abs(RCS_TE*a_vector_sum(2*MAX+2:4*MAX+2)).^2);
end
length_cy=20*ind_x; % Length of cylinder for approximately calculating the RCS of finite cylinders
RCS=10.*log10(RCS*2*sin(theta)*(length_cy)^2/ind_x);%
plot(phi,RCS)
xlabel('Phi')
ylabel('RCS (dB)')
hold on
%% Cylindrical harmonic functions
% notice i and j
% Cylindrical Bessel 1 (J_n)
function output=cb1(degree,k_rho,rho)
output=besselj(degree,k_rho*rho);
% Derivative of Cylindrical Bessel 1 (J_n')
% function output=cb_der1(degree,k_rho,rho)
% output=(degree*cb1(degree-1,k_rho,rho)-(degree+1)*cb1(degree+1,k_rho,rho))/(2*degree+1);
function output=cb_der1(degree,k_rho,rho)
output=(cb1(degree-1,k_rho,rho)-cb1(degree+1,k_rho,rho))/2;
%different from the book?j_{n-1}-n/(k*rho)*j_{n}%
% Cylindrical Bessel 2 (H^1_n)
function output=cb2(degree,k_rho,rho)
output=besselh(degree,1,k_rho*rho);
% Derivative of Cylindrical Bessel 2 (H^1_n')
% function output=cb_der2(degree,k_rho,rho)
% output=(degree*cb2(degree-1,k_rho,rho)-(degree+1)*cb2(degree+1,k_rho,rho))/(2*degree+1);
function output=cb_der2(degree,k_rho,rho)
output=(cb2(degree-1,k_rho,rho)-cb2(degree+1,k_rho,rho))/2;
% J matrix
function output=J_matrix(degree,k_rho,k_z,rho,eps,omega,mu)
output=1/(k_rho^2*rho)*[1j*omega*eps*k_rho*rho*cb_der1(degree,k_rho,rho),-1*degree*k_z*cb1(degree,k_rho,rho);
-1*degree*k_z*cb1(degree,k_rho,rho), -1j*omega*k_rho*rho*mu*cb_der1(degree,k_rho,rho) ];%
% H matrix
function output=H_matrix(degree,k_rho,k_z,rho,eps,omega,mu)
output=1/(k_rho^2*rho)*[1j*omega*eps*k_rho*rho*cb_der2(degree,k_rho,rho),-1*degree*k_z*cb2(degree,k_rho,rho);
-1*degree*k_z*cb2(degree,k_rho,rho), -1j*omega*mu*k_rho*rho*cb_der2(degree,k_rho,rho) ];
% modified J and H matrix for the calculation of rho component of E field
% modified J matrix
function output=J_matrix2(degree,k_rho,k_z,rho,eps,omega,mu)
output=1/(k_rho^2*rho)*[omega*eps*degree*cb1(degree,k_rho,rho),1j*k_z*k_rho*rho*cb_der1(degree,k_rho,rho);
1j*k_z*k_rho*rho*cb_der1(degree,k_rho,rho), -1*omega*mu*degree*cb1(degree,k_rho,rho) ];
% modified H matrix
function output=H_matrix2(degree,k_rho,k_z,rho,eps,omega,mu)
output=1/(k_rho^2*rho)*[omega*eps*degree*cb2(degree,k_rho,rho),1j*k_z*k_rho*rho*cb_der2(degree,k_rho,rho);
1j*k_z*k_rho*rho*cb_der2(degree,k_rho,rho), -1*omega*mu*degree*cb2(degree,k_rho,rho) ];