Read on if you’re interested in an easy way to calculate power density – even in your head.
Laser engineers and technicians are often required to calculate a laser’s power density to determine whether a beam would damage an optic or sensor or for other various applications. By definition, power density is power per unit area which is usually expressed in terms of W/cm2.
The Calculation of power density is straightforward. By simply calculating the area of a beam using the radius in cm and dividing the beam’s power by that area the power density in units of W/cm2 is easily obtained. However, since beam size is usually given with the beam diameter in terms of millimeters to calculate power density in W/cm2 a user has to convert the diameter to cm, divide the diameter by two to find the radius, use πr2 to find the area in cm2, and finally divide the laser power by the area to obtain power density. This calculation can be a bit tedious and time consuming especially for technicians and field engineers who want to make as quick and easy a calculation as possible.
With the following formula you can directly find power density of a laser beam using the diameter of the beam in millimeters:
Here is how this equation is derived: We can write an expression for the power density of a1 mmdiameter beam, which is simply:
Dividing the expression of power density of a 1 mmbeam — Power / π(0.5mm)2 — by an expression of power density as a function of diameter — Power / π(0.5d)2 — we find that the ratio is d2. Thus, power density as a function of diameter can be expressed as the power density of a 1 mm beam divided by d2:
This formula assumes the beam is a flat top beam where the beam power density is uniform. For a Gaussian beam with a given beam waist in mm, multiply this formula by two to obtain the formula given above (The multiplication factor is due to the fact that for a Gaussian beam, the peak power in the center is twice the average power density of the beam. The value is actually closer to 255, not 250, but this difference is trivial, introducing only ~2% error. We use 250 simply because it’s easier to remember and perform mental calculations than 255! )
Ophir’s Photonics group manufactures, calibrates and sells a complete line of accurate NIST compliant instruments for analyzing and measuring laser power, energy, position, size and beam profile (Spiricon LLC & Photon brands), which are used in industrial, medical and scientific research.
Ophir Photonics group is part of MKS Instruments, Inc., a global provider of technologies that enable advanced processes and improve productivity.