= Jones-component ratio sensitivity
{c}
{title2=$r'(0)=[J'_y(0)J_x(0)-J_y(0)J'_x(0)]/J_x(0)^2$}
For a differentiable <Jones vector> with $J_x(0)\ne0$, 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 $\mathbf J'(0)=a\mathbf J(0)$ for a complex scalar $a$, then $r'(0)=0$. For a lossless normalized output, differentiation of $\mathbf J^\dagger\mathbf J=1$ makes $\operatorname{Re}a=0$, so this change is a common <phase>. A vanishing component requires the complementary ratio or a basis-independent <polarization> coordinate instead.
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