Capacitive current division in heterodyne receivers

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 givesThe characteristic frequency $f_c=1/(2\pi R{C}_d)$ marks amplitude $1/\sqrt{2}$, 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, $f_c\approx 3.183$ 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 correctionThe correction also carries calibration uncertainty. Its logarithmic sensitivity to capacitance isAt $x=10$, 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.^[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.

Capacitance uncertainty in heterodyne current recovery is the contribution of uncertain shunt capacitance to the inferred generated beat electric current. In the capacitive current division in heterodyne receivers model, a measured load-current amplitude $A$ is corrected towith load electrical resistance $R$, angular frequency $\omega$ and shunt capacitance $C$. These amplitudes must refer to the stated current-plane convention in heterodyne calibration and use the same peak or root mean square convention.

Differentiation gives the logarithmic sensitivity coefficientThus capacitance contributes little when $x\ll 1$, contributes half its fractional change at $x=1$, and contributes nearly its full fractional change when $x\gg 1$. High-frequency correction increases the required amplitude estimate; it does not make uncertainty in the correction disappear.

Let $y=(\log A,\log \omega ,\log R,\log C{)}^{\mathsf{T}}$ have covariance matrix ${\Sigma }_y$. First-order uncertainty propagation givesThis form retains correlations rather than silently assigning four independent inputs. It is the GUM's sensitivity-and-covariance law applied to this particular measurement model.^[ref]

For an original small-error example, set $x=1$, neglect frequency uncertainty, and assign fractional standard uncertainties of $2\%$ to $A$, $1\%$ to $R$ and $5\%$ to $C$. With independent inputs,If only the $R,C$ errors have correlation coefficient $+0.8$, the result becomes about $3.54\%$; correlation $-0.8$ instead gives about $2.92\%$. The same marginal uncertainties can therefore produce different output uncertainty.

Nonlinearity also shifts the mean. Holding $A,\omega ,R$ fixed and writing $C=C_0+ϵ$ with centered small error, a second-order Taylor series givesAt $x_0=1$ and $u(C)/C_0=0.05$, this is about $0.0313\%$, distinct from the first-order spread. Broad or asymmetric inputs require a fuller distribution model.

A proposed diagnostic characterizes the load and shunt response at the actual bias and relevant beat frequencies, then propagates the joint calibration record. NIST's primary apparatus separately calibrates electrical networks and mismatches as well as controlling optical beating.^[ref] A fitted capacitance is only one part of that response: a single-pole model should not absorb untested cable, transit-time or amplifier effects.