I want to understand the meaning of the source code of S parameter analysis group shown below.

This is for extraction of effective index and impedance for metamaterials.

Analysis/Script/Low 209-211 & 212-214

```
n1 = (-1i*log( x+1i*sqrt(1-x^2) ) ) / (k*d);
n2 = (-1i*log( x-1i*sqrt(1-x^2) ) ) / (k*d);
n = (imag(n1) >= 0)*n1 + (imag(n1) < 0)*n2;
z1 = sqrt( ( (1+S11)^2-S21^2 ) / ( (1-S11)^2-S21^2 ) );
z2 = -sqrt( ( (1+S11)^2-S21^2 ) / ( (1-S11)^2-S21^2 ) );
z = (imag(z1) <= 0)*z1 + (imag(z1) > 0)*z2;
```

Why do we have to separately consider n1 and n2, or z1 and z2, with different symbol of â€śsqrtâ€ť in each equation? Why do we separately calculate â€śimag(n1) >= 0â€ť and â€śimag(n1) <0â€ť in the equations of n, and also z?

Thanks in advance.