 Gain-matching condition for heterodyne normalization

The 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 $I_1,I_2$, thenExact cancellation therefore requires $g_B^2=g_1{g}_2$, or equivalentlyThe 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 $1/0.99\approx 1.01010$. 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.^[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.