 Pulse-contrast ratio denominator noise

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 givesThis 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.^[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. ButThe second-order prediction is 0.0101 because $\mathrm{Var}(C)=1$. 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 $S=r_{0}C$ 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 butThis 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.^[ref] Resolve the denominator and background experimentally before interpreting a long average of ratios.