= Capacitive current division in heterodyne receivers
\b[Capacitive current division in heterodyne receivers] occurs when part of a <photodiode>'s generated alternating current flows through shunt capacitance instead of reaching the measurement load. It changes both amplitude and phase. A DC arm-current measurement does not experience the same frequency-dependent division, so the electrical response must be included when checking the <gain-matching condition for heterodyne normalization>.
In an original lumped-element model, a generated peak-current phasor $I_d$ feeds a capacitance $C_d$ in parallel with a resistive load $R$. Current balance gives
$$
H(\omega)=\frac{I_L}{I_d}=\frac1{1+i\omega RC_d},\qquad
|H|=\frac1{\sqrt{1+(\omega RC_d)^2}},\qquad
\arg H=-\arctan(\omega RC_d).
$$
The characteristic frequency $f_c=1/(2\pi RC_d)$ marks amplitude $1/\sqrt2$, phase $-\pi/4$ and delivered power one half of its low-frequency value for the same generated current. This is a current-divider model; a real receiver can have additional transit-time, amplifier and transmission-line responses.
For stipulated $R=50\,\Omega$ and $C_d=1$ pF, $f_c\approx3.183$ GHz. At 1 GHz the amplitude ratio is approximately 0.9540; at 10 GHz it is approximately 0.3033. Those values describe the stated equivalent circuit, not a particular detector specification.
Recovering generated amplitude requires the correction
$$
|I_d|=|I_L|\sqrt{1+x^2},\qquad x=\omega RC_d.
$$
The correction also carries calibration uncertainty. Its logarithmic sensitivity to capacitance is
$$
\frac{\partial\log|I_d|}{\partial\log C_d}=\frac{x^2}{1+x^2}.
$$
At $x=10$, a small 5% capacitance uncertainty contributes approximately 4.95% relative amplitude uncertainty if other inputs are fixed. Dividing by an accurately known response cannot recover signal-to-noise lost before a fixed downstream noise source; calibration and noise are different issues.
<Capacitance uncertainty in heterodyne current recovery> should be evaluated at the actual detector bias and frequency model. A proposed diagnostic sweeps beat frequency while holding optical conditions stable, compares magnitude and phase, and separately characterizes load and network response. A fitted single pole is evidence about that circuit approximation, not proof that every observed roll-off is capacitive.
NIST's primary receiver apparatus uses a tunable two-laser beat and separately measures scattering parameters, reflection coefficients, power-sensor calibration and interconnecting-network corrections.<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} Those measurements help distinguish detector response from cable or mismatch effects. Retain the <current-plane convention in heterodyne calibration> when comparing the measured load current with generated photocurrent.
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