Gain-matching condition for heterodyne normalization ($g_B^2=g_1{g}_2$)

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.

A heterodyne RF-to-DC gain transfer check tests whether the calibration converting a beat-frequency measurement into current amplitude is compatible with the calibration of the separate arm currents. The gain-matching condition for heterodyne normalization involves amplitude gains, so an RF power gain cannot be substituted without conversion.

For corrected peak beat current $B$ and arm currents $I_1,I_2$, the normalized overlap is $\gamma =B/(2\sqrt{I_1{I}_2})$. Suppose an electrical model gives sinusoidal peak current $B$ through a known resistive load $Z$. Its average beat power isSource/load division and impedance corrections must already be included when relating that load current to the detector-current convention. A factor converting power is therefore square-rooted when converting amplitude.

Let measured RF power carry multiplicative factor $g_P$, while the arm currents carry factors $g_1,g_2$. Under this model,Cancellation requires $g_P=g_1{g}_2$. With one common DC current factor $g_{\mathrm{DC}}$, the condition is $g_P=g_{\mathrm{DC}}^2$, not $g_P=g_{\mathrm{DC}}$.

An original example takes $g_{\mathrm{DC}}=1.10$. An RF power factor 1.21 produces amplitude factor 1.10 and cancels in the normalized overlap. Applying RF power factor 1.10 instead leaves multiplier $1/\sqrt{1.10}\approx 0.95346$, despite using the same numerical gain label in both paths. The example isolates heterodyne amplitude-versus-power gain conversion, rather than a change of optical overlap.

NIST's receiver experiment measured microwave power and separately corrected sensor calibration, electrical loss and impedance matching. Its two-laser optical source used polarization and spatial matching to establish calculable modulation.^[ref] Those separate operations make the RF-to-DC comparison a physical calibration problem.

Record peak versus RMS conventions, frequency response, load model and arm-current calibration. Test a known or independently characterized modulation while varying power within the linear range. Changed polarization, aperture clipping or optical alignment can alter the true overlap and imitate a gain-transfer defect, so one unexplained normalized ratio does not uniquely diagnose the electrical path.

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 $R$,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 $10{\log }_{10}4=20{\log }_{10}2\approx 6.0206$ dB. NIST distinguishes the factor ten for power ratios from the factor twenty for field-amplitude ratios when their quadratic relationship applies.^[ref]

If the load resistance also changes, the more general relation isFor 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 peak amplitude $B_d$, the normalized overlap is $\gamma =B_d/(2\sqrt{I_1{I}_2})$. If the numerator is instead detector-current RMS amplitude, the denominator is $\sqrt{2I_1{I}_2}$. Using an RMS numerator with the peak denominator lowers the result by $1/\sqrt{2}$, 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 $B_d^{2}R/8$, rather than $B_d^{2}R/2$. 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.^[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.

A current-plane convention in heterodyne calibration specifies which electrical current an inferred beat amplitude denotes: the detector's generated small-signal current, the current entering a cable, or the current dissipating power in a particular load. Those amplitudes need not be equal. A power-to-current conversion is meaningful only with its circuit and reference plane.

An original photodiode model uses generated peak-current phasor $I_d$, detector shunt admittance $Y_d$ and load impedance $Z_L$. Kirchhoff's current balance givesThe factor one half assumes peak phasors; with RMS phasors it disappears. The current in the load determines delivered electric power. The generated current is recovered only after correcting the current division and any intervening network.

For a resistive Norton model with source resistance and load resistance both $R$, $Y_d=1/R$, so $I_L=I_d/2$ and $P_L=|I_d{|}^{2}R/8$. If the source shunt admittance is negligible, $I_L\approx I_d$ instead. Calling both circuits “a detector feeding a 50-ohm instrument” does not establish the same current conversion.

Capacitive current division in heterodyne receivers adds frequency dependence. With $Y_d=i\omega C_d$ and resistive load $R$,For stipulated $C_d=1$ pF and $R=50\Omega$, this magnitude is approximately 0.9540 at 1 GHz and 0.3033 at 10 GHz. The same generated amplitude then gives a smaller measured load current at the higher frequency. These are equivalent-circuit calculations, not measured specifications of a particular photodiode.

NIST's primary optical-receiver experiment distinguishes microwave power delivered to a reference load from the power indicated by a sensor. It corrects sensor calibration, interconnecting-network loss and impedance mismatch using measured scattering and reflection parameters.^[ref] That apparatus makes the reference plane an experimental calibration choice rather than a label applied after analysis.

Document the detector model, load, frequency, cable and connector corrections, and peak-versus-root mean square convention before normalizing by separate arm currents. If the arm currents describe generated DC photocurrent while the beat numerator describes uncorrected load current, the resulting overlap includes electrical current division. Heterodyne amplitude-versus-power gain conversion and impedance matching address different parts of that chain and both may be needed.

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.