Saddle-point optical transition contour ($k_x^2-k_y^2=(\hslash \omega -E_s)/A$)

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 $d{k}_y/[2A\sqrt{k_y^2+\delta /A}]$. The gradient vanishes at the saddle, so this is a limiting description as nonzero detuning approaches zero.

For positive detuning inside a strict square region $|k_x|,|k_y|\le \Lambda$, substituting $k_x^2=k_y^2+\delta /A$ bounds $k_y$ by the displayed value, provided $0<\delta /A<{\Lambda }^2$. Integrating to $\Lambda$ instead extends the contour slightly outside that square. Either prescription has the same leading logarithmic singularity near zero detuning; it does not give the same exact finite-detuning result.