= Inversion of a symmetric Gaussian overlap log-ratio
{title2=$\tanh u=-\tanh(D_h/4)/\tanh h$}
\b[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 $D_h=-4\operatorname{artanh}(\tanh h\tanh u)$, define
$$
z=-\frac{\tanh(D_h/4)}{\tanh h}.
$$
Then
$$
u=\operatorname{artanh}z,\qquad
\rho=e^u=\sqrt{\frac{1+z}{1-z}}.
$$
A 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 $100\,\mathrm{\mu m}$, the inferred second radius is $50\,\mathrm{\mu m}$. 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 is
$$
\frac{du}{dD_h}
=-\frac{1-\tanh^2(D_h/4)}{4\tanh h(1-z^2)}.
$$
For small errors, relative radius uncertainty is approximately $u(\rho)/\rho\simeq|du/dD_h|u(D_h)$. 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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2]{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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements]{ref} Those controls motivate checks on the inversion assumptions; the paper is not a report of the particular symmetric-radius example derived here.
Back to article page
