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Write $r=\sqrt{\mu /\delta }$. At first order the amplitude-phase perturbation is $A=r+\epsilon (A_1+ir\theta )+O({\epsilon }^2)$. Expanding the cubic term in the focusing nonlinear Schrodinger equation gives $A|A{|}^2=r^3+\epsilon (3r^2{A}_1+i{r}^3\theta )+O({\epsilon }^2)$. The constant terms cancel because $\delta r^2=\mu$. Separating real and imaginary parts yieldsFor the Fourier normal modes, these equations give $-i\Omega {\hat{A}}_1=-\chi r{K}^2\hat{\theta }$ and $-ir\Omega \hat{\theta }+(2\mu -\chi K^2){\hat{A}}_1=0$. Their determinant vanishes exactly whenThe focusing nonlinear Schrodinger modulation dispersion is negative for $0<K^2<2\mu /\chi$. One of the two frequencies then has positive imaginary part and grows exponentially: the constant state has modulational instability. The growth rate is $\sqrt{\chi K^2(2\mu -\chi K^2)}$, maximized at $K^2=\mu /\chi$ with value $\mu$. Modes beyond the unstable band have real frequencies. The zero-wavenumber phase mode is neutral, with possible linear phase drift from a uniform amplitude perturbation.