= Symmetric log-overlap ratio
{title2=$D_h=\log(V_+/V_-)$}
A \b[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 readings
$$
V_+=A\operatorname{sech}^2(u+h),\qquad
V_-=A\operatorname{sech}^2(u-h),\qquad A>0.
$$
Here $q$ and $1/q$ multiply the second radius. The readings are efficiencies proportional to squared field overlap, not field amplitudes. The shared nuisance scale cancels in
$$
D_h=\log(V_+/V_-)=-2\log\frac{\cosh(u+h)}{\cosh(u-h)}.
$$
Using the addition formula for <hyperbolic cosine>, put $b=\tanh h$ and $z=\tanh u$ to obtain
$$
D_h=-4\operatorname{artanh}(bz).
$$
Thus $D_h$ 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:
$$
\frac{\partial D_h}{\partial u}
=-\frac{4b(1-\tanh^2u)}{1-b^2\tanh^2u}<0.
$$
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 $V_-=0.7\cdot64/289\approx0.15502$. Their log-ratio is $\log(289/64)\approx1.50754$, independent of the chosen scale. The allowed bound is $4\log2\approx2.77259$. 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements]{ref} The displayed relations are model derivations whose use requires fixed displacement, <wavefront curvature> and polarization across both settings.
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