= 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 leaves
$$
e(P)=\delta(P)-L(P).
$$
This 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,
$$
e(P)=\tfrac12\delta''(\xi)(P-P_1)(P-P_2)
$$
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 $|\delta''|\le M$, then
$$
|e(P)|\le\frac M2(P-P_1)(P_2-P)
\le\frac{M(P_2-P_1)^2}{8}.
$$
The 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 $M=0.02\,\mathrm{mW^{-2}}$. The midpoint error is bounded by $0.01$ in log-ratio. The exact function
$$
\delta(P)=L(P)+0.01\,\mathrm{mW^{-2}}(P-P_1)(P-P_2)
$$
attains $e(2\,\mathrm{mW})=-0.01$ while matching both endpoint corrections perfectly. The corresponding multiplicative error in the corrected optical ratio is $e^{-0.01}\approx0.99005$. Repeating endpoint measurements indefinitely would reduce their random uncertainty without revealing this midpoint discrepancy.
To guarantee $|e|\le\epsilon$ throughout a reference interval under the stated bound, choose spacing satisfying
$$
P_2-P_1\le\sqrt{\frac{8\epsilon}{M}}.
$$
The numerical bound $M=0.02\,\mathrm{mW^{-2}}$ and tolerance $\epsilon=0.0025$ 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p.1, Heterodyne measurements, mode matching]{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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2, higher-order terms]{ref}
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