How to Calculate Power Density Easily - Even in Your Head

Laser engineers and technicians are often required to calculate the power density of a laser. This is required for various applications, among other things to decide whether a laser beam would destroy an optic or a sensor. For definition: Power density is the power per unit area, it is usually given in W/cm².

The calculation of the power density is actually a straightforward matter: you divide the laser power by the area of ​​the beam in cm² and you get the result with the correct unit W/cm². However, the beam size is normally specified as a beam diameter in mm, so that the user must first convert this size into cm. Division by two then gives the radius, from which the area in cm² can be calculated using the formula πr². Only now can the beam power be divided by the area calculated in this way in order to obtain the power density. Due to the conversion of the units, this method of calculation harbors the risk of miscalculation. Especially when the field service needs to do the math in their heads as quickly as possible.

With the following formula you get directly the power density of the laser beam using the beam diameter in mm.

Power density Gauss (W/cm²) = 250/d² x power

You can find the derivation of the formula here: For the power density of a laser beam with a diameter of 1 mm, we can write down the following expression:

Power density (W/cm²) = power/ πr² = power/π(0.5 mm)² ≈ 127 x power (W/cm²)

Now dividing this expression as the power density of a 1mm beam - power/π(0.5mm)² - by an expression of power density as a function of diameter - power//π(0.5d)² - the result is the square of the diameter d². Consequently, the power density as a function of diameter can also be expressed as the power density of a 1mm beam divided by d²

Power density flat top (W/cm²) ≈ 127/ d² x power (W/cm²)

This formula assumes the beam is a flat top beam where the power density is uniform. For a Gaussian beam with a given beam diameter d in mm, the formula must be multiplied by two (The multiplier ² results from the fact that the peak power at the center of a Gaussian beam is twice the average power density of the beam). The value is actually closer to 255 and not 250; but the difference is marginal, causing an error of only about 2%. On the other hand, 250 is easier to remember and easier to calculate in your head.