Pulse energy

Pulse energy is the total optical energy in one pulse of a pulsed laser, the integral of the instantaneous power over the pulse. It is quoted in joules and, in practice, mostly in sub-multiples: nanojoules for mode-locked oscillators, microjoules for Q-switched fiber and solid-state lasers at tens to hundreds of kilohertz, and millijoules to joules for low-repetition-rate Q-switched lasers and amplified systems. A 2 W mode-locked oscillator at 80 MHz delivers 25 nJ per pulse; a 10 W Q-switched laser at 100 kHz delivers 100 µJ, 4000 times more energy per pulse at five times the average power.

Relation to average power

For a train of identical pulses at repetition rate $f_\text{rep}$, the average power is the energy per pulse times the number of pulses per second, so

$$E = \frac{P_\text{avg}}{f_\text{rep}} .$$

The examples above follow directly: $2\ \text{W} / 80\ \text{MHz} = 25$ nJ and $10\ \text{W} / 100\ \text{kHz} = 100$ µJ. An amplifier built on chirped-pulse amplification producing 1 mJ at 1 kHz has an average power of only 1 W, which is why the average power reading alone says little about what a pulsed laser does to a sample. The CW vs pulsed lasers entry compares the two regimes. For a single shot, or a burst, the energy is a property of each pulse and the average power has no fixed meaning.

Peak power and fluence

Dividing the energy by the pulse duration $\tau$ (intensity FWHM) gives the peak power, with a factor $k$ that depends on the pulse shape:

$$P_\text{peak} = k\,\frac{E}{\tau} ,$$

with $k = 0.881$ for a sech² pulse and 0.939 for a Gaussian. The 25 nJ oscillator pulse at 100 fs (sech²) peaks at about 220 kW; the 100 µJ pulse at 10 ns (Gaussian) at about 9.4 kW, lower despite the larger energy.

Dividing the energy by an area gives the fluence, the quantity that governs ablation and damage with nanosecond pulses. For a Gaussian beam of $1/e^2$ radius $w$, the peak fluence at the beam center is

$$F_0 = \frac{2E}{\pi w^{2}} .$$

The 100 µJ pulse focused to a spot of 50 µm $1/e^2$ diameter ($w$ = 25 µm) gives $F_0$ = 10.2 J/cm². Dividing instead by $\pi w^2$ gives the average fluence over the $1/e^2$ disk, 5.1 J/cm², a factor of two lower; a quoted fluence is only meaningful together with the area definition behind it.

Measuring pulse energy

At repetition rates up to the kilohertz range (some heads reach tens of kilohertz), single pulses are measured with an energy meter based on a pyroelectric detector: the absorbing element heats during the pulse, and the peak of its output voltage is proportional to the energy. These meters resolve each pulse, so they give the pulse-to-pulse distribution as well as the mean. At higher repetition rates the pyroelectric element cannot recover between pulses, and the energy is obtained by measuring the average power with a thermal or photodiode optical power meter and dividing by the repetition rate measured with a fast photodiode and counter. A thermopile power meter averages over its response time, typically seconds, and is indifferent to the pulse structure, which makes it robust for this purpose, provided the head tolerates the peak fluence.

Fast photodiodes follow individual pulses at megahertz rates but saturate at high peak power, so they serve for relative monitoring of pulse-to-pulse fluctuations.

Pulse-to-pulse stability

Pulse energy is rarely perfectly constant. Its stability is usually quoted as the RMS deviation in percent over a stated number of pulses, and Q-switched lasers in particular vary more as the repetition rate rises, because the gain medium has less time to recover between pulses; the energy per pulse also falls at high rates, as described under Q-switching. In threshold or nonlinear processes, such as ablation or harmonic generation, a small energy fluctuation produces a larger fluctuation in the result, so stability can matter as much as the mean.

Pitfalls

Amplified spontaneous emission, leakage between pulses and pre-lasing contribute to the average power but not to the useful pulse, so $P_\text{avg}/f_\text{rep}$ overstates the energy in such lasers. When pulse picking or burst mode is in use, the repetition rate to divide by is the delivered rate after picking, which is lower than the oscillator's.

Common questions

How do I convert average power to pulse energy?

Divide the average power by the repetition rate: 1 W at 1 kHz is 1 mJ, and 1 W at 80 MHz is 12.5 nJ. This assumes all the average power is in the pulses.

Is a higher pulse energy the same as a higher peak power?

No. Peak power is the energy divided by the duration, so a 25 nJ femtosecond pulse can exceed the peak power of a 100 µJ nanosecond pulse, as in the examples above.

How many photons are in a pulse?

The energy divided by the photon energy $hc/\lambda$: a 1 µJ pulse at 1064 nm contains about $5.4 \times 10^{12}$ photons.

References: A. E. Siegman, Lasers (University Science Books, 1986); W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).