Power density

Power density is the optical power crossing or falling on a unit area, $P/A$, quoted in laser work in W/cm² (1 W/cm² = 10⁴ W/m²). It is the same physical quantity as irradiance, the radiometric name, and as intensity in the sense used in laser physics and nonlinear optics; "power density" is the informal engineering name, common in data sheets, safety discussions and materials processing. Typical values span many orders of magnitude: about 2 W/cm² at the peak of a 5 mW helium-neon beam of 0.4 mm radius, about 1.3 MW/cm² for 1 W focused to a 10 µm spot, and about 320 MW/cm² for 1 kW in the 20 µm core of a fiber laser.

Naming

In radiometry, intensity means power per unit solid angle (W/sr), so "intensity" alone is ambiguous outside laser work. In electronics and signal analysis, "power density" often means the power spectral density, power per unit bandwidth in W/Hz or dBm/Hz, which has nothing to do with area. And in thermal design, the heat flux through a chip or submount is also called power density, in W/cm², though it describes heat flow. The units usually settle which is meant.

Gaussian beams: peak and average

For a uniform (top-hat) beam, power density is simply power divided by area. A Gaussian beam of power $P$ and $1/e^2$ radius $w$ has a profile $I(r) = I_0\exp(-2r^2/w^2)$, and its peak is

$$I_0 = \frac{2P}{\pi w^2},$$

twice the value obtained by dividing the power by the $1/e^2$ area $\pi w^2$. Taking 1 W focused to a spot 10 µm in diameter, so $w$ = 5 µm = 5 × 10⁻⁴ cm:

$$\frac{P}{\pi w^2} = \frac{1\ \text{W}}{7.85\times 10^{-7}\ \text{cm}^2}$$ $$\approx 1.27\times 10^{6}\ \text{W/cm}^2,$$

and the on-axis peak is 2.55 MW/cm². A third number, the mean power density inside the $1/e^2$ circle, is lower again: only 86.5% of the power falls inside radius $w$, so the mean there is 1.10 MW/cm². Quoting "1.3 MW/cm²" without saying which convention was used leaves a factor of two of ambiguity, which matters for damage and nonlinear thresholds, both of which respond to the peak.

Measurement

Power density is almost always derived rather than measured directly. The power comes from an optical power meter and the beam size from a camera beam profiler or a knife-edge scan at the plane of interest, usually the beam waist; for a Gaussian beam the peak follows from the formula above, and for other profiles a camera image normalized to the measured power gives a map of W/cm². For pulsed lasers the energy per area is the fluence, and the peak power density is the peak power divided by the area, or the fluence divided by the pulse duration for a roughly rectangular pulse.

Damage and facet limits

Optical damage of coatings, crystals and semiconductor facets depends on power density (or fluence for short pulses), so the same laser can be harmless in a 5 mm beam and destructive at a focus. Laser diodes are the clearest case: at the output facet of a GaAs-based emitter, power densities of order 1–10 MW/cm² on an unprotected facet start the thermal runaway called catastrophic optical damage. Spread over a 100 µm × 1 µm broad-area aperture, 12 W is 12 MW/cm², which is why high-power emitters use wide stripes and passivated facets. Continuous-wave damage specifications for optics are often given instead as a linear power density in W/cm, the power divided by the beam diameter, because for a CW beam the temperature rise at the center scales with $P/w$ rather than $P/w^2$. One watt in a 10 µm diameter spot is 1000 W/cm in that measure, and the two numbers cannot be compared directly.

Volumetric power density

Inside a gain medium the relevant quantity is often the power deposited per unit volume, in W/cm³, which drives temperature gradients and thermal lensing. In a thin-disk laser with an absorbed pump of 4 kW/cm², of which about 10% becomes heat, the 0.4 kW/cm² heat flux deposited in a 200 µm disk corresponds to an average heat density of 20 kW/cm³, which the thin geometry lets escape through the cooled back face.

Common questions

Is power density the same as intensity?

In laser physics and nonlinear optics, yes: both mean optical power per unit area, in W/cm². In radiometry, intensity means power per unit solid angle, and in signal analysis power density often means power per hertz, so the context and units decide.

How is the power density of a laser beam calculated?

Divide the power by the beam area. For a Gaussian beam, the peak is $2P/(\pi w^2)$ with $w$ the $1/e^2$ radius at the plane of interest. A 1 W beam focused to $w$ = 5 µm reaches about 2.5 MW/cm² at the center.

What is the difference between power density and fluence?

Power density is a rate, W/cm²; fluence is the energy per unit area delivered by a pulse or exposure, J/cm². For a pulse of duration $\tau$, fluence is roughly power density times $\tau$.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); A. E. Siegman, Lasers (University Science Books, 1986); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).