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A symmetric log-overlap ratio compares two measured Gaussian-beam overlap efficiencies after equal and opposite logarithmic changes of one radius. It removes a common multiplicative response scale and retains the sign of the initial radius mismatch.

In the centered, matched-curvature model, let $\rho =w_2/w_1$, $u=\log \rho$ and $h=\log q>0$. The two controlled settings have positive efficiency readingsHere $q$ and $1/q$ multiply the second radius. The readings are efficiencies proportional to squared field overlap, not field amplitudes. The shared nuisance scale cancels inUsing the addition formula for hyperbolic cosine, put $b=\tanh h$ and $z=\tanh u$ to obtainThus $D_h$ is odd in $u$, positive when $\rho <1$ and negative when $\rho >1$. Its finite-radius range is $|D_h|<4h$. As $u$ tends to either infinity, the ratio approaches a limiting value even though individual efficiencies can become very small.

The derivative makes the sensitivity explicit:At matching it is $-4\tanh h$; far from matching it approaches zero. A saturated log-ratio consequently supplies little information about additional radius mismatch.

For an original example, take $q=2$, $\rho =1/2$ and $A=0.7$. The positive setting gives $V_{+}=0.7$, while the negative setting gives $V_{-}=0.7\cdot 64/289\approx 0.15502$. Their log-ratio is $\log (289/64)\approx 1.50754$, independent of the chosen scale. The allowed bound is $4\log 2\approx 2.77259$. Taking logarithms of field-amplitude readings instead would give half this log-ratio.

A symmetric-overlap normalization drift breaks the simple cancellation: different scales $A_{+},A_{-}$ add $\log (A_{+}/A_{-})$ to the ideal $D_h$. Interleave the settings and monitor conditions to test that sharing assumption. NIST's receiver apparatus explicitly controlled spatial mode and polarization, but did not establish this particular symmetric-radius diagnostic.^[ref] The displayed relations are model derivations whose use requires fixed displacement, wavefront curvature and polarization across both settings.