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Boundary-sensitive Gaussian overlap inversion describes the loss of stable radius information when a symmetric log-overlap ratio approaches its finite-model limit. A central reading can remain formally invertible while small measurement uncertainty reaches values for which no finite radius ratio exists.

For known $h=\log q>0$, define $D=D_h$, $u=\log \rho$ andThe inversion of a symmetric Gaussian overlap log-ratio requires $|D|<4h$. At the upper limit $D\to 4h$, the radius ratio tends to zero; at the lower limit it tends to infinity.

An original expansion near the upper limit puts $ϵ=4h-D>0$. Expanding the hyperbolic tangent givesThus the approximate relative radius uncertainty is $u(\rho )/\rho \simeq u(D)/(2ϵ)$ when linearization is still justified. The distance to the boundary, rather than the magnitude of $D$ alone, controls this sensitivity.

For stipulated $q=2$ and a central log-ratio only 0.01 below $4\log 2$, the asymptotic radius ratio is about 0.0365. A standard uncertainty 0.02 in $D$ is twice the boundary distance, so a small symmetric error bar on the inferred radius is not credible. In a hypothetical Gaussian readout model centered there, roughly 31% of readings fall beyond the finite-inversion limit. This is a statistical illustration, not an observed receiver distribution.

Interval propagation for bounded overlap data shows the asymmetry directly. A stated interval in $D$ that reaches the upper bound maps to radius ratios arbitrarily close to zero; it does not yield a symmetric finite relative interval. An interval wholly inside the domain maps monotonically, with its endpoint order reversed.

Discarding out-of-range Monte Carlo draws would condition on a selected subset and change the distribution being propagated. Instead, examine the forward measurement model, admissible parameter range and calibration uncertainty, or construct an explicitly stated inference using the measured data. JCGM 101 separates input-distribution assignment from numerical propagation and states conditions for valid Monte Carlo use.^[ref]

Finally, proximity to the mathematical boundary is not the only possible problem. A change in shared normalization, polarization or wavefront curvature can violate the one-radius model. NIST's receiver apparatus illustrates the need for optical matching and separate electrical corrections.^[ref] Numerical clipping cannot replace those physical checks.

Inversion of a symmetric Gaussian overlap log-ratio recovers a radius ratio from two positive overlap-efficiency measurements made at known symmetric radius settings. It applies to the one-parameter model of symmetric beam-radius perturbation for overlap branch selection, with one shared response scale and fixed remaining mode properties.

Let $h=\log q>0$, $D_h=\log (V_{+}/V_{-})$ and $u=\log \rho$, where $\rho =w_2/w_1$. Since the symmetric log-overlap ratio satisfies $D_h=-4\mathrm{artanh}(\tanh h\tanh u)$, defineThenA finite positive radius ratio exists exactly when $|z|<1$, equivalently $|D_h|<4h$. The strictly decreasing forward relation makes the solution unique inside this interval. Equality corresponds only to a limiting zero or infinite radius ratio; an out-of-range noisy measurement has no finite solution in this ideal model.

For an original example, $q=2$ and $D_h=\log (289/64)$ give $\tanh h=0.6$ and $z=-0.6$, hence $\rho =1/2$. If the known first radius is $100\mu \mathrm{m}$, the inferred second radius is $50\mu \mathrm{m}$. The nuisance scale is unnecessary for that inversion. Once $\rho$ is obtained, one can reconstruct it as $A=V_{+}{\cosh }^2(u+h)$, but two fitted readings alone do not validate the Gaussian optical model.

The inverse sensitivity isFor small errors, relative radius uncertainty is approximately $u(\rho )/\rho \simeq |du/d{D}_h|u(D_h)$. At the worked point the derivative is $-17/30$; stipulated $u(D_h)=0.02$ gives about 1.13% relative radius uncertainty, before including uncertainty in the setting $h$ or the reference radius.

Near the allowed interval's edges, the inverse becomes highly sensitive. Boundary-sensitive Gaussian overlap inversion needs an uncertainty model that respects the nonlinear transformation; clipping an out-of-range value to a convenient radius does not supply the missing information. The GUM distinguishes local uncertainty propagation from cases requiring higher-order treatment.^[ref]

The apparatus must implement a radius change without unintended displacement, polarization or wavefront changes. NIST's physical heterodyne receiver experiment used mode and polarization control and separate electrical corrections.^[ref] Those controls motivate checks on the inversion assumptions; the paper is not a report of the particular symmetric-radius example derived here.

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.

A symmetric beam-radius perturbation for overlap branch selection compares two controlled settings of one Gaussian beam radius: $q{w}_2$ and $w_2/q$, with known $q>1$. Under the centred, matched-wavefront curvature model of reciprocal size-ratio ambiguity in centred Gaussian overlap, writeThe hyperbolic secant form makes the symmetry explicit: the two settings evaluate the same even function at $u+h$ and $u-h$. Their ordering identifies the sign of $u$, as shown in branch selection by a controlled beam-radius change.

An original extension removes an unknown common scale and recovers the radius ratio itself. Suppose the two measured positive efficiencies satisfyThe factor $A$ is a shared multiplicative nuisance parameter, such as an unchanged response normalization. The symmetric log-overlap ratio eliminates it exactly:For the last equality, use the addition formula for hyperbolic cosine to write the fraction as $(1+t)/(1-t)$, where $t=\tanh u\tanh h$. The defining logarithmic identity for the inverse hyperbolic tangent completes the derivation. Because $h>0$, $D_h$ is positive for the initially narrower beam, negative for the wider beam, and zero at matching.

The inversion of a symmetric Gaussian overlap log-ratio now givesA finite radius ratio exists exactly when $|D_h|<4h$. This condition follows from $|\tanh u|<1$; equality is approached only as one radius ratio tends to zero or infinity. A measured log-ratio outside that interval has no finite solution in this ideal two-setting model.

For an original numerical example, take $q=1.5$, true $\rho =0.8$ and unknown scale $A=0.75$. The ideal readings are $V_{+}=0.7256114$ and $V_{-}=0.5172352$. They give $D_h=0.3385169$, recovering $\rho =0.8$ without supplying $A$ to the inversion. The physical bound is $4h=1.6218604$.

Finite measurement noise still limits branch selection. For small relative errors, the delta method givesA perfectly shared multiplicative fluctuation cancels sample by sample, paralleling common-gain cancellation in heterodyne overlap uncertainty. Independent relative standard uncertainties of $1\%$ in the two readings instead give $u(D_h)\simeq 0.014142$. At the worked point, the derivative of the inverse gives $du/d{D}_h=-0.6780385$, hence about $0.959\%$ relative standard uncertainty in $\rho$. At exact matching the two branches coincide; a noisy sign cannot identify a distinct physical branch there.

These conclusions require $A$ and the other optical mode matching conditions to remain shared across the settings. Changing a lens can alter wavefront curvature and polarization as well as radius. NIST's physical heterodyne apparatus explicitly controlled polarization and spatial matching.^[ref] Larger radius changes may strengthen the log-ratio response while lowering either measured efficiency and changing its noise. Step choice therefore depends on the apparatus and noise model.

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.