Optimization of complex function

Hello,

After solving a linear equation to find a transmission coefficient for dialectric structure I get the next function -

T_coeff_opt = @(x) (256*(cos(2*(pi)*(2*x(1) + x(2))) - sin(2*(pi)*(2*x(1) + x(2)))*(1i)))/(- 144*cos(2*(pi)*(8*x(1)*(1i) - x(2)*(1i))*(1i)) + 400*cos(2*(pi)*(8*x(1)*(1i) + x(2)*(1i))*(1i)) - 306*sin(2*(pi)*(8*x(1)*(1i) - x(2)*(1i))*(1i))*(1i) + 850*sin(2*(pi)*(8*x(1)*(1i) + x(2)*(1i))*(1i))*(1i) + 900*sin(2*(pi)*x(2))*(1i));

Where x(1), x(2) are the thicknesses of the structure.

I want to optimize the function so I can get the thicknesses x(1), x(2) that satisfy:

1, Maximum transmission (that is defined [T_coeff_opt*conj(T_coeff_opt)].

2, I want to do that for a range of [0, 2*pi].

So for every phase point between [0, 2*pi] i'll get x(1), x(2) that gives me the correct phase with maximum transmission.

I've tried to do that with LSQNONLIN but the results are not what I expected. So maybe there is some other function that handles complex functions better.

I'm using for initial guass for w(n) [n>2] the previous solution to create a smooth (continuous) solution without many jumps or discontinuities.

Also the solution should be periodic [2*pi period].

My code goes like this:

startpoint = 1;
endpoint = 360;
dev = 360;
x_p = linspace(-1,1,dev);
T_phase = (pi)*x_p;
T_coeff_opt = @(x) (256*(cos(2*(pi)*(2*x(1) + x(2))) - sin(2*(pi)*(2*x(1) + x(2)))*(1i)))/(- 144*cos(2*(pi)*(8*x(1)*(1i) - x(2)*(1i))*(1i)) + 400*cos(2*(pi)*(8*x(1)*(1i) + x(2)*(1i))*(1i)) - 306*sin(2*(pi)*(8*x(1)*(1i) - x(2)*(1i))*(1i))*(1i) + 850*sin(2*(pi)*(8*x(1)*(1i) + x(2)*(1i))*(1i))*(1i) + 900*sin(2*(pi)*x(2))*(1i));
x0a = [0.4 0.6]; % Staring initial
lb = [0.08 0.08]; % Lower bound of width
ub = [1 1]; % Upper bound of width
options.Algorithm = 'trust-region-reflective';
options.StepTolerance = 1e-6;
options.FunctionTolerance = 1e-30;
% options = optimoptions('lsqnonlin','Display','off');
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective', 'Display', 'off');
F = @(x)Opt(x,startpoint,T_phase);
[w(startpoint,:), resnorm(startpoint,:), residual(startpoint,:)] = lsqnonlin(F, x0a, lb,ub,options);
for i = startpoint+1:1:endpoint
    F = @(x)Opt(x,i,T_phase);
    counter = 0;
    [w(i,:), resnorm(i,:), residual(i,:)] = lsqnonlin(F, w(i-1,:),lb,ub,options);
    minRes = residual(i,:);
    while (residual(i,:) > 1e-6 | counter < 10) % Trying to find a better solution.
        r = 0.0001*rand(1,1);
        [curr_w, resnorm(i,:), residual(i,:)] = lsqnonlin(F, r*w(i-1,:),lb,ub,options);
        if (residual(i,:) < minRes)
            minRes = residual(i,:);
            w(i,:) = curr_w;
        end
        counter = counter + 1;
    end 
end

function F = Opt(x,i, T_phase)
     T_coeff_opt = (256*(cos(2*(pi)*(2*x(1) + x(2))) - sin(2*(pi)*(2*x(1) + x(2)))*(1i)))/(- 144*cos(2*(pi)*(8*x(1)*(1i) - x(2)*(1i))*(1i)) + 400*cos(2*(pi)*(8*x(1)*(1i) + x(2)*(1i))*(1i)) - 306*sin(2*(pi)*(8*x(1)*(1i) - x(2)*(1i))*(1i))*(1i) + 850*sin(2*(pi)*(8*x(1)*(1i) + x(2)*(1i))*(1i))*(1i) + 900*sin(2*(pi)*x(2))*(1i));
    F(1) = (((T_coeff_opt)*conj(T_coeff_opt)) - 1);
    F(2) = (angle(T_coeff_opt) - (T_phase(i)));
end

Thanks for any help!