Pulse energy fluctuations are usually described using a root-mean-square (rms) definition: https://www.rp-photonics.com/article_noise_in_laser_technology2.html

And yes, this is the same as the relative standard deviation.

Pulse-to-pulse stability (ptp) shows average deviation of pulse energy for adjacent pulses in pulse sequence. Standard deviation (std) shows average deviation of pulse energy from average (for total sequence) pulse energy. That`s why ptp can be better or worse than std. It depends on the process, where we consider these characteristics. To calculate ptp you must replace [E(i) -E(average)] to [E(i)-E(i+1)] in the formula for std.

Hello.

As can be found in the scientific literature, there is no standard methodology for estimating the pulse-to-pulse stability of a laser. Most manufacturers prefer to operate the laser for a reasonable time and analyze frequency, temperature, and power stability in the form of a time series of collected data. Also, others prefer to observe the average power stability by measuring the amplitude fluctuations in a successive collection of pulses to infer about the energy stability, taking into account that the amplitude is related to the energy of the pulse. Although these techniques only give us some information regarding pulse stability, they can provide a quantitative measurement of pulse-to-pulse energy fluctuations.

On the other hand, the statistical approximation allows us to estimate the stability in terms of the Standard Deviation (SD) of the histograms, so the lower the SD, the greater the energy stability (this is the relationship between Relative SD and pulse-to-pulse stability).

The results can also be related to the laser parameters, for example in a Nd: YAG rod-Laser, the parameters used to characterize the laser are the density of active ions in the laser rod, the spontaneous decay lifetime, the absorption coefficient, the volume and the length of the rod, the physical size of the cavity, the photon density "𝜙" at the frequency "𝜐" and the fractional photon losses "𝛾" (DC and AC losses due to output coupling and saturable absorber). But as can be inferred, everything will depend on the geometry of the Laser and the parameters mentioned, among others.

Ref. "Pulse-to-pulse stability analysis in a frequency-doubled, q-switched Nd:YAG rod-Laser". Matheus A. Tunes, Cláudio G. Schön, Niklaus Ursus Wetter