= Pulse-contrast ratio denominator noise
\b[Pulse-contrast ratio denominator noise] is the distortion introduced when a noisy central pulse signal is used to normalize a side-delay signal. A <shot-resolved pulse-contrast estimator> uses $r=S/C$ after background correction, so errors in $C$ affect both precision and the distribution of the ratio.
For central values $\mu_S,\mu_C$ with $\mu_C>0$, a second-order local expansion gives
$$
\mathbb E\!\left[\frac SC\right]\simeq
\frac{\mu_S}{\mu_C}
+\frac{\mu_S\operatorname{Var}(C)}{\mu_C^3}
-\frac{\operatorname{cov}(S,C)}{\mu_C^2}.
$$
This approximation requires sufficiently small fluctuations and well-behaved ratio moments. It shows why an unbiased central-signal measurement need not give an unbiased normalized ratio. First-order <uncertainty propagation> alone describes local spread, not this mean shift; the GUM warns that nonlinear models can require higher-order terms.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2, nonlinear functions; §5.2, correlated inputs]{ref}
An original bounded example takes constant side signal $S=0.1$ and equally likely central readings $C=9$ or 11. The mean central reading is 10, so the ratio of means is 0.01. But
$$
\mathbb E[S/C]=\frac{0.1}{2}\left(\frac19+\frac1{11}\right)=\frac1{99}\approx0.010101.
$$
The second-order prediction is 0.0101 because $\operatorname{Var}(C)=1$. Increasing the number of such shots estimates the shifted mean more precisely; it does not remove the nonlinear effect.
Correlation can instead make normalization exact. If $S=r_0C$ for every accepted shot with nonzero $C$, their ratio is always $r_0$, however variable the common multiplier. This is a different error structure from adding independent noise to the two channels. A shared additive background is also not generally equivalent to a shared multiplicative gain.
Near zero, even existence of a mean can fail. If $S=s_0>0$ is fixed and $C$ is uniform on $(0,1)$, every ratio is positive but
$$
\mathbb E[S/C]=s_0\int_0^1\frac{dc}{c}=\infty.
$$
This mathematical example explains why “the denominator was never exactly zero” is insufficient. Detector noise is not asserted to follow this distribution.
A <pulse-contrast denominator acceptance rule> such as $C\ge C_{\min}$ changes the sampled population. Report the rule, rejected-shot count and dependence between brightness and pulse shape. The primary single-shot correlator's nonlinear gate and CCD readout motivate acquiring the relevant signals together, but do not guarantee that all ratios are statistically well conditioned.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printed pp. 194–195, Description and Fig. 2]{ref} Resolve the denominator and background experimentally before interpreting a long average of ratios.
Back to article page
