Symmetric-overlap normalization drift

Symmetric-overlap normalization drift is a time-dependent change in the multiplicative scale of two overlap readings. The symmetric log-overlap ratio cancels one shared scale, but sequential settings need not experience that same scale.

Let the stable optical contributions at the two radius settings be $F_{+}$ and $F_{-}$, and letwhere $a(t)=\log A(t)$. The ideal log-ratio is $D_0=F_{+}-F_{-}$. A positive-setting reading at time 0 followed by a negative-setting reading at time $\Delta$ instead gives $D_{\mathrm{obs}}=D_0+a(0)-a(\Delta )$. A drift in normalization can therefore mimic a signed radius mismatch.

An original four-reading design uses settings $+,-,-,+$ at equally spaced times $0,\Delta ,2\Delta ,3\Delta$. FormThe scale contribution vanishes exactly when $a(t)$ is linear in time: both settings have the same mean observation time. This cancels linear drift in logarithmic scale, not arbitrary drift in every measured quantity.

If $a(t)=a_0+bt+c{t}^2$, the remaining drift contribution is $2c{\Delta }^2$. For stipulated $D_0=0.05$, $b=0.02$ per time unit and $\Delta =1$, a simple two-reading comparison gives 0.03 under purely linear log-drift, while the four-reading design gives 0.05. Adding $c=0.003$ per squared time unit changes the latter to 0.056. These are analytical timing examples, not measurements from a beam experiment.

With independent equal log-readout variance ${\sigma }^2$, the four-reading estimator has variance ${\sigma }^2$, because it combines four coefficients of magnitude one half. A single two-reading difference has variance $2{\sigma }^2$ at half the acquisition count. Correlated noise and longer acquisition time can change the practical tradeoff.

State-dependent gain in symmetric overlap measurements does not cancel through this ordering. If the scale changes systematically with the radius-setting state, it remains confounded with $F_{+}-F_{-}$. The optical mismatch must also remain stable over the sequence; a changing beam is not merely an electronic scale drift.

NIST controlled mode and polarization matching while calibrating the receiver's electrical response.^[ref] The ordering above is an original drift-control calculation, not an attributed NIST procedure. Test its assumptions with timing records, repeated sequences and independent monitoring before feeding the resulting ratio into the radius inversion.

State-dependent gain in symmetric overlap measurements is a multiplicative response that changes with the optical radius-setting state. It can imitate the radius mismatch inferred from a symmetric log-overlap ratio. An acquisition order that cancels time drift does not remove a scale change tied to the state itself.

For positive overlap readings, supposewhere $u=\log \rho$ is the true logarithmic radius ratio, $A_s$ is a fixed state-specific gain and $a(t)$ is shared logarithmic drift. Even after perfectly removing $a(t)$, the measured log-ratio isHere $h=\log q>0$ describes the known symmetric radius settings. The two unknowns $u$ and $\delta$ cannot be determined from this one ratio without further information. This is an identifiability problem, not merely a larger readout variance.

For an original matched-beam example, $\rho =1$ gives $D_h=0$. A state-gain ratio $A_{+}/A_{-}=1.04$ instead produces $D_{\mathrm{obs}}=\log 1.04\approx 0.03922$. With $q=2$, ignoring the gain difference givesapproximately 0.98379. The apparently small radius mismatch is entirely instrumental in this stipulated example. Repeating the same states more often reduces independent scatter without removing the state offset.

A reference-state transfer in overlap normalization can supply additional information. If a reference has known $u=0$ and the same state-gain ratio as the specimen measurement, its observed log-ratio estimates $\delta$. Subtracting that log-ratio is justified only if the state response transfers. Changing optical power, detector operating point or aperture illumination between reference and specimen can break that assumption.

Another explicit model uses a simultaneous monitor $M_s=K{A}_s{e}^{a(t)}$ with the same constant $K$ in both states. Then $\log V_s-\log M_s=F_s(u)-\log K$, and the state difference cancels the monitored gain. A separate optical-power monitor does not automatically satisfy this equation for an RF-amplifier gain change: verify which part of the measurement chain the monitor actually samples.

NIST's primary receiver apparatus uses variable attenuators, a beam splitter, a polarizer and single-mode-fibre coupling to match the incident modes, followed by electrical calibration.^[ref] Those controls motivate distinguishing an electronic gain change from an optical change in polarization, clipping or beam shape. The reference and monitor designs above are original diagnostic models, not procedures attributed to that paper.

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]