Erf function in complex domain problems

Question

Dear all,

form one month i'm looking for an erf function in the complex domani to be able to plot a complex function representing cavity ringdown in  a micro resonator.

This is my script:
%% Cavity ringdown function
% Valutazione dei parametri del cavity ringdown a partire da dati
% sperimentali
%
% Biblio: 
% - "Determination of coupling regime of high-Q
% resonators and optical gain of highly selective
% amplifiers", Dumeige et al. Vol. 25, No. 12/December 2008/J. Opt. Soc. Am. B
%%
clear
%% Experimental data
c_light = 3e8;
% Quality factor
% Q = omega_risonanza * tau/2
% Q = freq_risonanza/delta_freq

% Valori presi da articolo Feron 2008 per testare il nostro script matlab
Q = 1*10^8;
Qe = 2*Q;
Q0 = 2*Q;


% Resonance wavelenght and frequency
lambda0 = 1.55*10^(-6);
omega0 = 2*pi*c_light/lambda0;
delta_omega0 = omega0/Q;
% Total photon lifetime tau/2
tau = 2*Q/omega0;
tau_e = 2*tau;
tau_0 = tau_e;
% Velocita' di riferimento, e' la velocita' con cui in un tempo tau viene
% percorsa la FWHM, infatti:
% v0_t = delta_12/(tau/2) = (1/pi*tau)/(tau/2) = 2/(pi*tau^2)
V0_t = 2/pi/(tau)^2;

% HWHM
delta_12 = omega0/(2*pi)/2/Q;
% Definita come sotto rappresenta la FWHM della risonanza, infaffi delta_12
% e' la HWHM, quindi FWHM = 2*HWHM, se moltiplico per 2pi ottengo la
% larghezza in frequenza angolare
% OMEGA = 2*pi*2*delta_12;
OMEGA = delta_omega0;
% Ts_ref rappresenta il Tempo con cui viene percorsa la HWHM (delta_12) con velocita' V0_t
Ts_ref = tau/2;

% Scala per asse dei tempi t/tau
t_scale = [400,20,8,4];
% Vs is the frequency sweeping speed
% Experimentally Vs is about [3 GHz /5 ms]
% Velocita' di scansione riferite a V0_t
A_Vs = [0.0075,0.3,3,30];
Vs_vect = A_Vs.*V0_t*2*pi;

n_points = 1*10^(3);
t = zeros(length(t_scale),n_points);
T = zeros(length(t_scale),n_points);
X = zeros(length(t_scale),n_points);


for iiii = 1:length(t_scale)
% tempo
t(iiii,:) = linspace(0,t_scale(iiii)*tau,n_points);
Vs = Vs_vect(iiii);
% Larghezza scansione attorno alla risonanza omega0, la definisco un certo numero di volte la
% larghezza della riga, ma e' solo una scelta
% Ad esempio con una risonanza larga 2MHz, si effettua una scansione di 20
% MHz attorno 
A_OMEGA_scan = 10;

OMEGA_scan = A_OMEGA_scan*OMEGA;
% Frequenza di inizio scansione
omega_i = omega0 - OMEGA_scan/2;
% Ts
Ts = A_OMEGA_scan/A_Vs(iiii)*tau;
% Frequenza di scansione
% omega_t = omega_i + OMEGA_scan/(Ts).*t(iiii,:); 
omega_t = omega_i + Vs/(2).*t(iiii,:); 

% Initial detuning
deltai = -OMEGA_scan/(4*pi);  

% Excitation frequency
phi_t = omega_t.*t(iiii,:);
%% Temporal variation
S0 = 1;
S_int(iiii,:) = S0*exp(1i*phi_t);
%%
f_t(iiii,:) = -sqrt(1i*pi/(2*Vs)).*exp((-1i*(2*pi*deltai - 1i./tau).^2)./(2*Vs)).*erfz((1i/tau - 2*pi*deltai - Vs.*t(iiii,:))./(sqrt(2*1i*Vs)));

u_t(iiii,:) = sqrt(2./tau_e).*S0.*exp(1i*omega0*t(iiii,:) - t(iiii,:)/tau).*(f_t(iiii,:) - f_t(iiii,1) + 1./(1i*(omega_t - omega0) + 1./tau));
%%
S_out(iiii,:) = -S_int(iiii,:) + sqrt(2./tau_e)*u_t(iiii,:);
%% Transmission
T(iiii,:) = abs(S_out(iiii,:)./S0).^2;

x(iiii,:) = t(iiii,:)./tau;

end
%% Immagini

scrsz = get(0,'ScreenSize');

figure('Position',[0 33 scrsz(3) scrsz(4)*9/10]);
subplot 221
plot(x(1,:),T(1,:));
title(['Transmission function. Q = ' num2str(Q,'%10.2e') '; {\lambda} = ' num2str(lambda0*10^6) ...
    ' {\mu}m; T_s = ' num2str(Ts*10^9) ' ns; {	au} = ' num2str(tau*10^9) ' ns; {	au}_e = '  num2str(tau_e*10^9) ' ns']);
xlabel('t/{	au}');
ylabel('T');
grid on;
legend(['Vs/V0 = ' num2str(A_Vs(1))]);


subplot 222
plot(x(2,:),T(2,:));
title(['Transmission function. Q = ' num2str(Q,'%10.2e') '; {\lambda} = ' num2str(lambda0*10^6) ...
    ' {\mu}m; T_s = ' num2str(Ts*10^9) ' ns; {	au} = ' num2str(tau*10^9) ' ns; {	au}_e = '  num2str(tau_e*10^9) ' ns']);
xlabel('t/{	au}');
ylabel('T');
grid on;
legend(['Vs/V0 = ' num2str(A_Vs(2))]);


subplot 223
plot(x(3,:),T(3,:));
title(['Transmission function. Q = ' num2str(Q,'%10.2e') '; {\lambda} = ' num2str(lambda0*10^6) ...
    ' {\mu}m; T_s = ' num2str(Ts*10^9) ' ns; {	au} = ' num2str(tau*10^9) ' ns; {	au}_e = '  num2str(tau_e*10^9) ' ns']);
xlabel('t/{	au}');
ylabel('T');
grid on;
legend(['Vs/V0 = ' num2str(A_Vs(3))]);

subplot 224
plot(x(4,:),T(4,:));
title(['Transmission function. Q = ' num2str(Q,'%10.2e') '; {\lambda} = ' num2str(lambda0*10^6) ...
    ' {\mu}m; T_s = ' num2str(Ts*10^9) ' ns; {	au} = ' num2str(tau*10^9) ' ns; {	au}_e = '  num2str(tau_e*10^9) ' ns']);
xlabel('t/{	au}');
ylabel('T');
grid on;
legend(['Vs/V0 = ' num2str(A_Vs(4))]);

%% Controllo dei NaN
disp(['T(NaN) = ' num2str(sum(sum(isnan(T(1,:)))))]);
disp(['f(t)(NaN) = ' num2str(sum(sum(isnan(f_t(1,:)))))]);
disp(['u(t)(NaN) = ' num2str(sum(sum(isnan(u_t(1,:)))))]);
disp(['Sin(NaN) = ' num2str(sum(sum(isnan(S_int(1,:)))))]);
disp(['Sout(NaN) = ' num2str(sum(sum(isnan(S_out(1,:)))))]);

%%%
However i have some problems both in plot than trying to fit experimental data with the "T" function.
I think some problems can come from the "erfz" function.
Can you give some suggestions?

All the best
Andrea

Erf function in complex domain problems

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