Jones-component ratio sensitivity ($r^{\prime }(0)=[J_y^{\prime }(0)J_x(0)-J_y(0)J_x^{\prime }(0)]/J_x(0{)}^2$)

For a differentiable Jones vector with $J_x(0)\ne 0$, the ratio $r=J_y/J_x$ fixes polarization independently of common amplitude and phase. The title quotient derivative tests first-order polarization sensitivity. If ${\mathbf{J}}^{\prime }(0)=a\mathbf{J}(0)$ for a complex scalar $a$, then $r^{\prime }(0)=0$. For a lossless normalized output, differentiation of ${\mathbf{J}}^{†}\mathbf{J}=1$ makes $\mathrm{Re}a=0$, so this change is a common phase. A vanishing component requires the complementary ratio or a basis-independent polarization coordinate instead.