Peak power

Peak power is the maximum of the instantaneous optical power $P(t)$ within a pulse. For a pulsed laser it is far larger than the average power a power meter reads, because the energy of each pulse arrives in a short interval and the laser is dark between pulses. A mode-locked Ti:sapphire oscillator averaging 1 W reaches about 110 kW at the peak of each 100 fs pulse; a 10 W Q-switched laser with 10 ns pulses at 100 kHz reaches about 9.4 kW. Peak power, and the peak intensity it produces at a focus, governs nonlinear optics, multiphoton absorption, material ablation and optical damage.

From average power to peak power

Two measured quantities give the pulse energy: the average power $P_\text{avg}$, from a thermal or photodiode power meter, and the repetition rate $f_\text{rep}$, from a photodiode and frequency counter:

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

The peak power then follows from the pulse duration $\tau$ (intensity FWHM) and a factor $k$ that depends on the pulse shape:

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

For a sech² pulse, integrating $P_0\,\mathrm{sech}^2(t/T_0)$ gives $E = 2P_0T_0$, and with $\tau = 1.763\,T_0$ the factor is $k = 1.763/2 = 0.881$. For a Gaussian, $E = \sqrt{\pi}\,P_0T_0$ with $\tau = 1.665\,T_0$, so $k = 1.665/\sqrt{\pi} = 0.939$. A rectangular pulse would have $k = 1$. The shape factor changes the result by about 6–12 %, which is often smaller than the uncertainty in the measured duration.

Worked examples

A Ti:sapphire oscillator with $P_\text{avg}$ = 1 W at 80 MHz has $E$ = 12.5 nJ. With 100 fs sech² pulses,

$$P_\text{peak} = 0.881 \times \frac{12.5\ \text{nJ}}{100\ \text{fs}} \approx 110\ \text{kW},$$

or 117 kW if the pulse is taken as Gaussian. The duty cycle, $\tau f_\text{rep}$, is 8 × 10⁻⁶, so the peak exceeds the average by a factor of about 10⁵.

A Q-switched laser with 10 W at 100 kHz has $E$ = 100 µJ. With 10 ns Gaussian pulses, $P_\text{peak}$ = 0.939 × 100 µJ / 10 ns ≈ 9.4 kW. The pulse energy is 8000 times larger than the oscillator's, but the peak power is lower, because the pulse is 10⁵ times longer.

Peak intensity at a focus

Nonlinear processes respond to the intensity at the focus. For a Gaussian beam focused to a waist radius $w_0$ (1/e² intensity radius), the on-axis peak intensity is

$$I_\text{peak} = \frac{2P_\text{peak}}{\pi w_0^{2}}.$$

The 110 kW oscillator pulse focused to $w_0$ = 10 µm gives $I_\text{peak} \approx 7.0 \times 10^{10}$ W/cm², and at $w_0$ = 2 µm, as under a high-numerical-aperture microscope objective, about $1.8 \times 10^{12}$ W/cm², enough to drive two-photon absorption efficiently. The 9.4 kW Q-switched pulse at $w_0$ = 25 µm gives about $9.6 \times 10^{8}$ W/cm², but its peak fluence, $2E/(\pi w_0^2)$, is 10.2 J/cm², against 0.008 J/cm² for the oscillator at 10 µm. For ablation and damage with nanosecond pulses fluence is usually the better predictor; for femtosecond nonlinear processes intensity is. The distinction between irradiance averaged over time and peak intensity within the pulse is a frequent source of errors of several orders of magnitude.

Where it matters

Peak power sets the strength of self-phase modulation, harmonic generation and self-focusing. Self-focusing in bulk glass becomes catastrophic above a critical power of a few megawatts for near-infrared light, which is the practical reason chirped-pulse amplification stretches pulses before amplifying them. In pulsed fiber lasers, peak power rather than average power usually limits scaling, through nonlinear effects in the small core. The pulsed laser calculator performs the conversions above for any combination of average power, repetition rate and duration.

Pitfalls

Using $E/\tau$ with a duration that was never deconvolved from an autocorrelation overstates the width by 41–54 % and understates the peak power by the same factor. Energy in a pedestal or satellite pulses counts in the average power but contributes little to the peak, so real peak power is lower than the formula predicts for poor pulses. Leakage from Q-switched lasers between pulses, or amplified spontaneous emission in amplifiers, similarly inflates the average power reading.

Common questions

How do you calculate peak power from average power?

Divide the average power by the repetition rate to get the pulse energy, then multiply the energy by 0.881 (sech²) or 0.939 (Gaussian) and divide by the pulse duration.

Why is peak power so much higher than average power?

Their ratio is approximately the inverse of the duty cycle, $1/(\tau f_\text{rep})$. For an ultrafast laser at 80 MHz with 100 fs pulses this is about 10⁵.

Can peak power damage a power meter?

A thermal power meter responds to average power, but its absorber coating can be damaged by high peak intensity or fluence even at modest average power, so manufacturers specify separate pulsed damage thresholds.

References: A. E. Siegman, Lasers (University Science Books, 1986); A. M. Weiner, Ultrafast Optics (Wiley, 2009); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006).