= State-dependent gain in symmetric overlap measurements
\b[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, suppose
$$
V_s(t)=A_s\exp(F_s(u)+a(t)),\qquad s\in\{+,-\},
$$
where $u=\log\rho$ 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 is
$$
D_{\rm obs}=D_h(u)+\delta,\qquad
\delta=\log(A_+/A_-),\qquad
D_h(u)=-4\operatorname{artanh}(\tanh u\tanh h).
$$
Here $h=\log q>0$ 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 $D_{\rm obs}=\log1.04\approx0.03922$. With $q=2$, ignoring the gain difference gives
$$
\rho_{\rm inferred}=\sqrt{\frac{1-z_0}{1+z_0}},\qquad
z_0=\frac{\tanh(D_{\rm obs}/4)}{\tanh(\log2)},
$$
approximately 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements]{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.
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