Inverse of Non-Linear Equations - MatLab - matlab

Question

I have been trying without success to figure out how to invert the two equations shown below. As you can see, the equations are fairly complex nonlinear equation expressing R and T (reflection and transmission coefficients in acoustics) as a function of the variables n, k, d, and xi. What I need is to invert these equations and get expressions for xi and n in terms of the other variables. i is the imaginary unit.
Starting Equations
I know the answer I need to get (second image) and I am trying to replicate this answer using some MatLab code. m is branch number of the cos inverse (arccos) function.
Solution to aim for

Answer 1

1.- Since surf, like plot and other commands, requires X and Y to be same length, graphically inverting the function, what in AutoCAD is called rotating the UCS rather that starting with a symbolic inversion attempt, makes sense.
UCS : Universal Coordinates System . The tripod on the CAD drawing sheet indicating axes and origin.
% MATLAB
close all;clear all;clc
c0=299792458; % [m/s] light velocity
% guessing frequency band centre
f0=2e9; % [Hz]
lambda0=c0/f0; % wavelength
k0=2*pi/lambda0 % wave number
d=.03 % 3 [cm]? chip size? connector? antenna? waveguide length? the kettle?
% range_n=[0:lambda0/10:lambda0/4];
range_n=[0:.01:4.25];
range_e=[0:.05:10];
[N,E]=meshgrid(range_n,range_e);
%% Reflection Coefficient
R=1j*tan(N*k0*d).*(1./E-E)./(2-1j*tan(N*k0*d).*(1./E+E));
figure(1);
ax1=gca;
hs1=surf(ax1,N,E,abs(R))
hs1.EdgeColor='none';
grid on;xlabel('n');ylabel('e');zlabel('abs(R)');title('|R(n,e)|')
xcr1_1=ax1.XTickLabel; % x axis x measure ruler
xcr1_2=ax1.YTickLabel; % x axis y measure ruler
figure(2);
ax2=gca;
hs2=surf(ax2,N,E,angle(R))
hs2.EdgeColor='none';
grid on;xlabel('n');ylabel('e');zlabel('angle(R)');title('phase(R(n,e)) [rad]')
%% Transmission Coefficient
T=2./(cos(N*k0*d).*(2-1j*tan(N*k0*d).*(1./E+E)));
figure(3);
ax3=gca;
hs3=surf(ax3,N,E,abs(T))
hs3.EdgeColor='none';
grid on;xlabel('n');ylabel('e');zlabel('abs(T)');title('|T(n,e)|')
It makes sense that where there's high reflection then there's low transmission and where there's low reflection, then T gets close or equal to 1.
It's a passive device / material.
figure(4);
ax4=gca;
hs4=surf(ax4,N,E,angle(T))
hs4.EdgeColor='none';
grid on;xlabel('n');ylabel('e');zlabel('angle(T)');title('phase(T(n,e))')
2.- Now E from the other side, the sought inversion :
E2up=abs((((1+R).^2-T.^2)./((1-R)-T.^2)).^.5);
E2down=-abs((((1+R).^2-T.^2)./((1-R)-T.^2)).^.5);
[sz_s11,sz_s12]=size(E2up);
range_s11=[1:1:sz_s11];
range_s12=[1:1:sz_s12];
[sz_s1,sz_s2]=size(E2up);
range_s11=[1:1:sz_s11];
range_s12=[1:1:sz_s12];
[S11,S12]=meshgrid(range_s11,range_s12);
% |E(S,T)|>0
figure(5);
ax5=gca;
hs5=surf(ax5,S11,S12,abs(E2up'));
hs5.EdgeColor='none';
grid on;xlabel('s1_R');ylabel('s1_T');zlabel('abs(E_u_p)');title('|E_u_p(s1_R,s1_T)|')
s1_R s1_T variables in this particular context ARE NOT scattering parameters.
s1_r s1_T are numerals, indexs related to R and T.
n and m already in use.
3.- Changing inverted surface X Y numerals to R T
ax5.XTickLabel=xcr1_1; % x axis x change ruler
ax5.YTickLabel=xcr1_2; % x axis y change ruler
xlabel(ax5,'R');ylabel(ax5,'T');zlabel(ax5,'|(E_u_p)| positive values only');
% |E(S,T)|<0
ax6=gca;
hs6=surf(ax6,S11,S12,abs(E2down'));
hs6.EdgeColor='none';
grid on;xlabel('s1_R');ylabel('s1_T');zlabel('abs(E2_d_o_w_n)');title('|E_d_o_w_n(s1_R,s2_T)|')
ax6.XTickLabel=xcr1_1; % x axis x change ruler
ax6.YTickLabel=xcr1_2; % x axis y change ruler
xlabel(ax6,'R');ylabel(ax6,'T');zlabel(ax6,'|(E_d_o_w_n)| negative values only');
4.- Comments :
4.1.- Repeat for angle(Edown(R,T)) and then for n_up(R,T) n_down(R,T).
4.2.- Including the 2*pi*m/(k0*d) shouldn't be difficult either.
4.3.- Assumed waveguide filled with air.
Dielectrics modify lambda0 to lambda k0 to k and may add more cut-offs.
4.4.- Is the material homogeneous and isotropic?