= Common-gain cancellation in heterodyne overlap uncertainty
\b[Common-gain cancellation in heterodyne overlap uncertainty] is the exact invariance of the normalized beat amplitude
$$
\widehat\gamma=\frac{B}{2\sqrt{I_1I_2}}
$$
when the corrected peak beat current $B$ and both separately measured arm currents $I_1,I_2$ are multiplied by the same factor $g>0$. The factor occurs once in the numerator and once in the denominator. This is <scale invariance>: the ratio is a <homogeneous function> of degree zero. The cancellation holds for every realization of a random common factor, without requiring it to be independent of the currents.
More generally, take $B,I_1,I_2>0$ and let their residual multiplicative <measurement calibration> factors be $g_B,g_1,g_2$. The observed normalization satisfies
$$
\frac{\widehat\gamma_{\mathrm{obs}}}{\widehat\gamma}
=\frac{g_B}{\sqrt{g_1g_2}}=e^d,
\qquad
d=\log g_B-\tfrac12\log g_1-\tfrac12\log g_2.
$$
Thus the <gain-matching condition for heterodyne normalization> is $g_B^2=g_1g_2$; the three factors need not all be equal. For example, $g_B=1$, $g_1=1.04$ and $g_2=1/1.04$ leave the ratio unchanged. The denominator depends on the <geometric mean> of its two factors.
Writing the <natural logarithms> of the factors as a <random vector> $\mathbf G$ makes the remaining uncertainty particularly clear. With $\mathbf v=(1,-1/2,-1/2)^T$,
$$
\operatorname{Var}(d)=\mathbf v^T\operatorname{Cov}(\mathbf G)\mathbf v.
$$
This identity is exact for the logarithmic error. The relative <variance> of the original ratio is only approximately equal to it when the errors are small, as in the <variance of a corrected heterodyne overlap estimate>. Keeping the <covariance> terms follows the general measurement-propagation rule.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
For an original imperfect-sharing model, set $\log g_j=\alpha_j Z+\epsilon_j$. Assume $Z,\epsilon_B,\epsilon_1,\epsilon_2$ are zero-mean <independent random variables>. The coefficient of the shared fluctuation is $c=\alpha_B-(\alpha_1+\alpha_2)/2$, so
$$
v_d:=\operatorname{Var}(d)
=c^2\operatorname{Var}(Z)+\operatorname{Var}(\epsilon_B)
+\tfrac14\operatorname{Var}(\epsilon_1)+\tfrac14\operatorname{Var}(\epsilon_2).
$$
Take $\alpha_B=1$, $\alpha_1=\alpha_2=0.95$, with <standard deviations> $0.03$ for $Z$, $0.01$ for $\epsilon_B$ and $0.004$ for each arm's separate error. Then $c=0.05$ and $v_d=0.00011025$, giving logarithmic <standard uncertainty> $0.0105$, approximately $1.05\%$ relative uncertainty. A large shared fluctuation can leave a small residual, but incomplete matching prevents exact cancellation.
If the log-errors additionally have a <normal distribution>, then $d\sim N(0,v_d)$ and the normalized readout has a <lognormal distribution>. Its exact <expected value> is $\gamma e^{v_d/2}$, not $\gamma$. At $\gamma=0.75$, this example gives mean $0.7500413$ and <standard deviation> $0.0078757$. The <lognormal bias of a heterodyne overlap estimate> comes from exponentiating the remaining error; the perfectly shared factor still cancels.
\Image[/heterodyne-gain-cancellation.png]
{title=A common scale cancels from normalized heterodyne overlap; a beat-channel-only scale does not}
{description=Original deterministic examples with nominal normalized overlap 0.75. The three curves apply the same scale to all currents, to the beat current alone, or with logarithmic sensitivity 1 for the beat and 0.95 for both arm currents. Independent readout noise is omitted to isolate the scaling algebra.}
The physical calibration paths determine which model applies. NIST's receiver experiment combined two lasers through a <beam splitter>, <polarizer> and <single-mode optical fiber>, while separately correcting electrical loss and impedance mismatch.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements and equations 1–4]{ref} An <electronic gain> change confined to the beat-frequency path need not appear in the arm-current measurements. Likewise, adding a current offset is not multiplication by a common factor. Background correction and the sharing of calibration factors must therefore be established before claiming cancellation.
Back to article page
