How it is calculated
The pulse energy, average power, and repetition rate are linked by
The peak power depends on how the energy is distributed in time. For a pulse of full width at half maximum \(\tau\),
The factor \(k\) is the ratio of the FWHM to the pulse's energy-equivalent width. A Gaussian's integral is \(1.0645\,P_\text{peak}\tau\). A sech² pulse has longer wings, and its integral is \(1.1346\,P_\text{peak}\tau\).
Smooth pulses have lower peaks, not higher
With the energy and FWHM fixed, the rectangular estimate \(E_p/\tau\) overestimates the true peak power, by 6.4 % for a Gaussian pulse and 13.5 % for sech². Smooth pulses spread some energy into wings outside the FWHM, so less is left for the peak. Factor-of-two differences in peak irradiance come from the beam's spatial profile: a Gaussian beam's on-axis fluence is twice the energy divided by the 1/e² area. The fluence and irradiance tool handles that.
| Shape | k = Ppeakτ/Ep | ∫P dt / (Ppeakτ) | Autocorrelation FWHM / τ |
|---|---|---|---|
| Rectangular | 1 | 1 | — |
| Gaussian | 0.9394 | 1.0645 | 1.414 |
| sech² | 0.8814 | 1.1346 | 1.543 |
The last column matters for ultrafast lasers. An intensity autocorrelator reports a width longer than the pulse, so divide by the deconvolution factor before using the duration here.
Worked examples
Q-switched fiber laser. 20 W average at 100 kHz with 10 ns Gaussian pulses:
The duty cycle is 0.1 %, so the peak power is about 940 times the average.
Ultrafast laser. 10 W at 1 MHz with 300 fs sech² pulses gives \(E_p = 10\ \mu\)J and \(P_\text{peak} = 0.8814 \times 10\ \mu\text{J}/300\ \text{fs} \approx 29\ \text{MW}\). The average power is modest, but the peak power is about three million times higher.
Assumptions and limits
- τ is the FWHM of the power envelope. Some datasheets quote 1/e or 10–90 % widths. Convert those before entering them.
- Real pulses are not symmetric. Q-switched pulses rise fast and decay slowly, and MOPA fiber lasers can be programmed to arbitrary shapes. If you have a measured waveform, integrate it and use \(k = P_\text{peak}\tau/E_p\).
- Pedestals and satellite pulses count toward the measured average power but contribute little to the peak. Amplified ultrafast systems can have percent-level ASE or pre-pulse content.
- Burst mode (groups of closely spaced pulses) is not modeled. Use the energy per sub-pulse and the burst envelope separately.
References
- A. E. Siegman, Lasers, University Science Books (1986), ch. 9 (pulse shapes and widths).
- J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed., Academic Press (2006), ch. 9 (autocorrelation deconvolution factors).