= Heterodyne amplitude-versus-power gain conversion
\b[Heterodyne amplitude-versus-power gain conversion] translates a calibration of measured <electric power> into a calibration of beat-current amplitude. In a fixed resistive load, power is quadratic in current amplitude, so a multiplicative power correction becomes a square-root amplitude correction. The electrical reference plane and peak-versus-<root mean square> convention must also agree.
For sinusoidal load current $i(t)=B\cos\omega t$ and resistance $R$,
$$
I_{\rm rms}=\frac B{\sqrt2},\qquad
P=RI_{\rm rms}^2=\frac{RB^2}{2}.
$$
For two measurements of the same load, $P_2/P_1=(B_2/B_1)^2$. A power multiplier of four therefore means an amplitude multiplier of two. In <decibels>, the same change is $10\log_{10}4=20\log_{10}2\approx6.0206$ dB. NIST distinguishes the factor ten for power ratios from the factor twenty for field-amplitude ratios when their quadratic relationship applies.<NIST Guide to the SI>[Chapter 8, §8.7, logarithmic quantities and units]{ref}
If the load resistance also changes, the more general relation is
$$
\frac{B_2}{B_1}=\sqrt{\frac{P_2R_1}{P_1R_2}}.
$$
For frequency-dependent complex impedances, use the appropriate circuit and power model rather than replacing resistance by an arbitrary complex number. A measured <electric potential difference> likewise requires its stated impedance before it can be converted to current.
For a corrected detector-current \i[peak] amplitude $B_d$, the normalized overlap is $\gamma=B_d/(2\sqrt{I_1I_2})$. If the numerator is instead detector-current RMS amplitude, the denominator is $\sqrt{2I_1I_2}$. Using an RMS numerator with the peak denominator lowers the result by $1/\sqrt2$, a 3.0103 dB discrepancy under the amplitude-ratio convention. The mistake is a convention mismatch, not evidence of worse optical overlap.
<Current-plane convention in heterodyne calibration> is another independent issue. In an ideal matched Norton circuit, a source peak current $B_d$ splits equally between source resistance and equal load resistance $R$. Load peak current is $B_d/2$, so delivered power is $B_d^2R/8$, rather than $B_d^2R/2$. The factor depends on the specified equivalent circuit; real detector capacitance and interconnecting networks require their measured response.
NIST's two-laser receiver apparatus measures microwave power and separately corrects sensor calibration, network loss and impedance mismatch. Its matched optical modes make the modulation calculable, while the electrical corrections locate the power at the intended reference plane.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements, Eqs. 1–4 and Fig. 1]{ref} Record raw power, correction factors, reference plane, load, amplitude convention and DC-current calibration separately. Only then apply the <gain-matching condition for heterodyne normalization>.
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