= Gain-matching condition for heterodyne normalization
{title2=$g_B^2=g_1g_2$}
The \b[gain-matching condition for heterodyne normalization] is the relation between three positive multiplicative calibration factors that leaves a normalized beat unchanged. If the corrected peak beat current is $B$ and the separate arm currents are $I_1,I_2$, then
$$
\widehat\gamma=\frac{B}{2\sqrt{I_1I_2}},\qquad
\frac{\widehat\gamma_{\rm obs}}{\widehat\gamma}
=\frac{g_B}{\sqrt{g_1g_2}}.
$$
Exact cancellation therefore requires $g_B^2=g_1g_2$, or equivalently
$$
\log g_B=\tfrac12(\log g_1+\log g_2).
$$
The beat-channel gain must equal the <geometric mean> of the arm-channel gains. The three gains need not individually agree.
An original example uses $g_1=1.21$ and $g_2=0.81$. Their geometric mean is 0.99, so $g_B=0.99$ gives exact cancellation. Keeping $g_B=1$ instead multiplies the inferred overlap by $1/0.99\approx1.01010$. Choosing the arithmetic mean 1.01 for $g_B$ would not solve the problem: it would multiply the result by approximately 1.02020. These are stipulated calibration factors, not receiver specifications.
For time-dependent gains, the condition must hold for the corresponding measurements in each realization if sample-by-sample invariance is claimed. Merely arranging zero mean logarithmic mismatch does not guarantee an unbiased arithmetic overlap: <lognormal bias of a heterodyne overlap estimate> gives a counterexample under Gaussian log-errors.
A <heterodyne RF-to-DC gain transfer check> distinguishes electronic amplification in the beat-frequency path from the response used for the arm currents. A change confined to $B$ does not cancel. Neither does a generic additive dark-current error satisfy this multiplicative model. Preserve the amplitude convention as well: $B$ is a corrected peak current, rather than the unconverted electrical beat power.
The NIST receiver experiment used two lasers, a <beam splitter>, a <polarizer> and single-mode collection to establish optical matching, while separately correcting the electrical measurement path.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements and Eqs. 1–4]{ref} Those different physical operations explain why one common scalar gain is a special case rather than an automatic description of the entire apparatus.
One practical test varies a suspected common gain while holding powers, mode matching and collection conditions fixed, and checks whether the normalized result stays constant. Passing that test constrains response to that change; it does not exclude an unchanged systematic mismatch. Report the gain-sharing model and the remaining uncertainty before interpreting normalization as complete calibration cancellation.
Back to article page
