Pulse timing jitter (${\sigma }_t$)

Pulse timing jitter describes variation of pulse arrival times relative to a stated timing reference. Choose a reproducible pulse feature and reference instants $t_j^{\mathrm{ref}}$; define errors $e_j=t_j-t_j^{\mathrm{ref}}$. One measure is their standard deviation ${\sigma }_t$. Edge-time error relative to an ideal reference differs from variation of successive periods or cycle-to-cycle differences; the primary timing guide distinguishes these measurements.^[ref]

For an original distribution, let each error equal $500\mathrm{ps}+ϵ$, where $ϵ$ is equally likely to be $+100$ or $-100\mathrm{ps}$. The mean delay is $500\mathrm{ps}$ and ${\sigma }_t=100\mathrm{ps}$. The root mean square error about zero is instead $\sqrt{500^2+100^2}\approx 510\mathrm{ps}$. A fixed delay changes the mean without increasing the spread. None of these numbers is the individual pulse's duration.

A fast photodetector, reference timing signal and oscilloscope can measure a selected arrival feature. A fixed electrical threshold can also turn amplitude/readout noise into an apparent time change, producing threshold timing noise from amplitude noise. Specify the reference, feature, observation duration and measurement response. A fitted or recovered reference can absorb some timing variation, changing the reported residual.

Timing variation can become gated-energy variation even at constant full pulse energy. The Gaussian gate centring error provides one quantitative model, distinct from pulse-to-pulse energy fluctuation of complete pulses.

Threshold timing noise from amplitude noise occurs when voltage fluctuations move the apparent time at which a pulse crosses a fixed threshold. Let a clean electrical trace $s(t)$ cross at $t_0$ with nonzero slope $S=s^{\prime }(t_0)$. For small additive noise $n$, linearizing $s(t_0+\delta t)+n=s(t_0)$ givesThe result concerns a selected measurement feature. It is not automatically the source's physical arrival-time spread. The timing primer distinguishes reference-edge error from other jitter statistics.^[ref]

For an original readout example, use noise standard deviation $2\mathrm{mV}$ and crossing slope $0.20V/ns$. The linear estimate is $10\mathrm{ps}$. At half that slope it becomes $20\mathrm{ps}$, even with the same noise amplitude. Both units and the local slope matter.

A pulse generator, receiver and oscilloscope can compare crossing times while the threshold position or signal slope changes. Small-noise linearization presumes a unique nearby crossing; ringing, large noise or multiple crossings require a fuller model. The relevant noise bandwidth must match the recorded trace.

A changing pulse scale can also shift a fixed threshold time even without additive noise. That amplitude-dependent time walk is a different effect from the random perturbation derived above. Pulse timing jitter describes the resulting timing observable, while the original optical arrival and detector readout contributions must be distinguished.