Optics / 03 / Pulses

Energy, rate,
and the peak.

Relate pulse energy, repetition rate and average power, and find the peak power for a stated pulse shape and duration.

Peak power depends on the shape. Energy divided by duration is only the peak of a flat-top pulse. Choose the shape that describes your pulse. Runs in your browser; nothing is sent or stored.

Pulse train

Temporal shape of the power P(t)

Gaussian sech² Rectangular

Duration τ, FWHM of the power

Full width at half maximum of the intensity, not of an autocorrelation trace.

Repetition rate f

Leave empty for a single pulse.

Known quantity

Pulse energy Average power

Pulse energy E

Result

Peak power P_peak: —
Average power: —
Shape factor k = P_peak τ / E: —
Pulse period 1/f: —
Duty factor f τ: —
Peak / average: —
Train peak including neighbours: —

Results to 4 significant figures. Displayed digits are not a measurement uncertainty.

Model and limits

P_{peak} = k \frac{E}{τ}

peak power; τ is the FWHM of the power P(t)

P_{avg} = E f_{rep}

average power

P (t) \propto \exp (- 4 \ln 2 \frac{t^{2}}{τ^{2}}), k = 2 \sqrt{\frac{\ln 2}{π}} \approx 0.9394

Gaussian pulse

P (t) \propto {sech}^{2} (\frac{1.7627 t}{τ}), k = \ln (1 + \sqrt{2}) \approx 0.8814

sech² pulse

k = 1

rectangular (flat-top) pulse of length τ

P_{train} = P_{peak} (1 + 2 \sum_{m \geq 1} p (\frac{m}{f_{rep} τ}))

peak of a train: the neighbours add their tails; p is the unit-peak shape

τ is the full width at half maximum of the optical power (intensity) in time. An intensity autocorrelation is wider by 1.414 (Gaussian) or 1.543 (sech²); divide by that factor first.
Every pulse in the train is identical and isolated. Satellite pulses, pedestals, ASE or a CW background carry energy without adding to the peak; the result is then an upper bound on the peak power.
P_peak is the peak of one isolated pulse. In a train the tails of the neighbours add at each peak: P_train = P_peak (1 + 2 Σ p(m / f τ)), with p the unit-peak shape. The sum is negligible at f τ = 0.1 (below 10⁻¹⁵) and reaches 0.003 % (Gaussian) or 0.7 % (sech²) at f τ = 0.5; it is shown when it exceeds 0.1 %. The page refuses f τ above 0.5 (Gaussian, sech²) or 1 (rectangular), where the train is close to continuous.
Peak power is not peak intensity: intensity also needs the beam area and spatial profile.

Sources

The shape factors follow from integrating each P(t): E = ∫P dt. For sech², ∫sech²(t/T) dt = 2T and τ = 2 acosh(√2) T.
J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006): pulse-shape and autocorrelation width factors.
Same shape factors as the LaserShark modules physics/amplifier.py and physics/damage.py.

Energy, rate,and the peak.

Pulse train

Result

Model and limits

Sources

Energy, rate,
and the peak.