Curvature error in overlap-reference transfer

Curvature error in overlap-reference transfer is the residual instrumental correction left when a curved operating-point response is represented by endpoint linear interpolation. Let $\delta (P)$ be the actual log-ratio offset at optical power $P$, and let $L(P)$ interpolate its exact values at $P_1<P_2$. Applying $L(P)$ as the correction leavesThis deterministic model discrepancy is separate from random uncertainty in the measured endpoints.

The polynomial interpolation error formula gives, for twice continuously differentiable $\delta$ and $P$ between the endpoints,for some $\xi$ in that interval. To derive it, subtract a multiple of $(P-P_1)(P-P_2)$ chosen to make the error vanish also at $P$; two applications of Rolle's theorem make the second derivative vanish at an intermediate point. If $|{\delta }^{\prime \prime }|\le M$, thenThe maximum product occurs at the midpoint. Halving the reference spacing therefore quarters this worst-case bound when the same curvature bound remains justified.

For an original example, use power in milliwatts, endpoints $P_1=1$, $P_2=3$, and $M=0.02\mathrm{m}{\mathrm{W}}^{-2}$. The midpoint error is bounded by $0.01$ in log-ratio. The exact functionattains $e(2\mathrm{mW})=-0.01$ while matching both endpoint corrections perfectly. The corresponding multiplicative error in the corrected optical ratio is $e^{-0.01}\approx 0.99005$. Repeating endpoint measurements indefinitely would reduce their random uncertainty without revealing this midpoint discrepancy.

To guarantee $|e|\le ϵ$ throughout a reference interval under the stated bound, choose spacing satisfyingThe numerical bound $M=0.02\mathrm{m}{\mathrm{W}}^{-2}$ and tolerance $ϵ=0.0025$ require spacing no greater than $1\mathrm{mW}$. Outside the endpoint interval, the product error formula needs a curvature bound over the larger range including the extrapolation point; the interior maximum bound cannot simply be reused.

A proposed check measures additional intermediate power settings and repeats at the intended detector bias and beat frequency. It also monitors mode alignment and polarization, which the primary NIST heterodyne apparatus controls separately.^[ref] Finitely many extra points constrain a response model but do not prove an arbitrary global curvature bound. State how $M$ is justified. The GUM's treatment of uncertainty propagation distinguishes local approximations from significant nonlinear effects.^[ref]