Jones vector ($\mathbf{J}=(J_x,J_y{)}^{\mathsf{T}}$)

A Jones vector records the two transverse complex amplitudes of a completely polarized monochromatic wave. Multiplication by a common nonzero complex number preserves its polarization of waves, while changing overall amplitude and phase. With the convention $\mathbf{\Psi }(t)=\mathrm{Re}[\mathbf{J}{e}^{i\omega t}]$, $\mathbf{J}=(1,2i{)}^{\mathsf{T}}$ gives $(\cos \omega t,-2\sin \omega t)$, an ellipse with the longer axis along $y$. Handedness labels require both a viewing direction and a naming convention.

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.