Reference-state transfer in overlap normalization

Reference-state transfer in overlap normalization uses a known optical state to estimate a state-dependent instrumental offset, then transfers that correction to the specimen measurement. Its central assumption is that the relevant state-gain ratio remains applicable between the reference and specimen. Repeatability of the reference alone does not establish transfer.

Let $u=\log \rho$, $h=\log q>0$ andSuppose specimen and reference observations satisfyWith known reference state $u_R$, the corrected specimen log-ratio isThe residual is exactly the failure of the instrumental offset to transfer. Acquisition order can address time drift, but cannot by itself prove ${\delta }_S={\delta }_R$.

An original matched-state example has $u_S=u_R=0$ and $q=2$. If the specimen's state-gain ratio differs from the reference's by $e^{0.02}$, the corrected ratio still contains 0.02. Ignoring that residual gives an inferred radius ratio approximately 0.9917 instead of one. Repeating both measurements can reduce independent scatter while preserving that systematic offset.

Reference uncertainty must also propagate. Near a matched reference, $D_h^{\prime }(0)=-4\tanh h$. For $q=2$, its magnitude is 2.4. A standard uncertainty 0.01 in the reference log-radius therefore contributes 0.024 to the corrected log-ratio before combining other errors. If the same reference estimate corrects many specimens, this uncertainty is shared; it does not become independent for each specimen. The GUM explicitly retains covariance from common references.^[ref]

Operating-point transfer of an overlap reference requires checking the variables capable of changing the state response: optical power, detector bias, electronic range, frequency and illumination. A useful proposed test brackets specimen readings with reference readings and repeats the reference at several operating powers. A trend then tests a transfer assumption; an unchanged reference at only one power leaves power-dependent transfer untested.

NIST's receiver apparatus controls spatial mode and polarization with a polarizer and single-mode-fibre coupling, while separately calibrating the electrical response.^[ref] A reference must preserve the relevant optical and electrical conditions, or the analysis must model their differences. The transfer calculation above is an original diagnostic construction, not a claimed NIST protocol.

Operating-point transfer of an overlap reference asks whether a state-dependent normalization correction measured under reference conditions remains valid at the specimen's optical and electrical settings. Detector bias, optical power, beat frequency, amplifier range and illumination can all define the operating point. Repeated readings at one setting establish repeatability there, rather than transfer over an untested range.

Let the true optical log-ratio be $D_h(u)$ and the instrumental state offset at power $P$ be $\delta (P)$. A reference measurement estimates $\delta (P_R)$. Applying it to a specimen at $P_S$ leavesThis follows directly from reference-state transfer in overlap normalization. The difference between operating points is a measurement-model input, not extra independent readout noise.

An original local model writes $\delta (P)=a+bP$. References at two powers $P_1<P_2$ support linear interpolation at an intermediate $P$:The weights sum to one. A specimen outside the tested range requires extrapolation, which is more sensitive to model error.

Curvature error in overlap-reference transfer is visible in an exactly solvable example. If the true offset is linear plus $c{P}^2$, its difference from endpoint interpolation is $c(P-P_1)(P-P_2)$. For powers 1 and 3 mW, suppose the measured endpoint offsets are 0.02 and 0.06. The interpolated offset at 2 mW is 0.04. A quadratic term $0.01(P-1)(P-3)$, with powers expressed in mW, leaves both endpoints unchanged but makes the actual midpoint offset 0.03. Correcting by 0.04 then leaves a log-ratio error of $-0.01$. These are stipulated response values, not observations.

Reference uncertainty also transfers. At a fixed, known interpolation power,Here $u_1,u_2$ are standard uncertainties; uncertainty in $P$ requires additional terms. A shared calibration can correlate the endpoints, so their covariance should not be dropped merely because they were measured at different powers.^[ref]

A useful proposed experiment brackets specimen acquisitions in time, samples several operating powers and repeats at the intended bias and frequency. Check for optical changes as well as electronic response: NIST's primary heterodyne apparatus controls spatial mode and polarization before electrical calibration.^[ref] The interpolation calculation above diagnoses a transfer assumption; it is not an attributed NIST procedure.

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]