= Saddle-point optical transition contour
{title2=$k_x^2-k_y^2=(\hbar\omega-E_s)/A$}
For a local transition-energy model $E_c-E_v=E_s+A(k_x^2-k_y^2)$ with $A>0$, positive detuning has two hyperbola branches opening along x. Negative detuning opens along y; the zero-detuning limiting contour is a pair of intersecting lines. The coarea integrand on one positive-detuning branch reduces to $dk_y/[2A\sqrt{k_y^2+\delta/A}]$. The gradient vanishes at the saddle, so this is a limiting description as nonzero detuning approaches zero.
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