= Symmetric beam-radius perturbation for overlap branch selection
A \b[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>, write
$$
u=\log\rho,\quad \rho=w_2/w_1,\quad h=\log q,
\qquad \eta(\rho)=\frac{4\rho^2}{(1+\rho^2)^2}=\operatorname{sech}^2u.
$$
The <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 satisfy
$$
V_+=A\operatorname{sech}^2(u+h),\qquad
V_-=A\operatorname{sech}^2(u-h),\qquad A>0.
$$
The factor $A$ is a shared multiplicative <nuisance parameter>, such as an unchanged response normalization. The <symmetric log-overlap ratio> eliminates it exactly:
$$
D_h=\log\frac{V_+}{V_-}
=-2\log\frac{\cosh(u+h)}{\cosh(u-h)}
=-4\operatorname{artanh}(\tanh u\tanh h).
$$
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 gives
$$
\boxed{\rho=\exp\left[\operatorname{artanh}\left(-\frac{\tanh(D_h/4)}{\tanh h}\right)\right].}
$$
A 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 $4h=1.6218604$.
\Image[/symmetric-gaussian-overlap-log-ratio.png]
{title=Two symmetric radius settings remove a shared scale and determine the signed Gaussian radius mismatch}
{description=Original exact model with radius multiplier 1.5 and common efficiency scale 0.75. The left panel shows the two efficiency readings; the right shows their scale-independent log-ratio, its limiting bounds and the worked radius ratio 0.8. All curves are analytical model predictions.}
Finite measurement noise still limits branch selection. For small relative errors, the <delta method> gives
$$
u^2(D_h)\simeq\frac{u^2(V_+)}{V_+^2}+\frac{u^2(V_-)}{V_-^2}
-\frac{2\operatorname{Cov}(V_+,V_-)}{V_+V_-}.
$$
A 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 $u(D_h)\simeq0.014142$. At the worked point, the <derivative> of the inverse gives $du/dD_h=-0.6780385$, 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements]{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.
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