# A shortcut for calculating Power Density of a laser beam

By Effy Shafner on Jul 5, 2012 in FAQ's

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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*/*cm ^{2}*.

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*/*cm ^{2}* is easily obtained. However, since beam size is usually given with the beam diameter in terms of millimeters to calculate power density in

*W*/

*cm*a user has to convert the diameter to cm, divide the diameter by two to find the radius, use

^{2}*πr*to find the area in

^{2}*cm*, 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.

^{2}

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*(0.5*mm*)* ^{2 }*– by an expression of power density as a function of diameter —

*Power*/

*π*(0.5

*d*)

*– we find that the ratio is*

^{2 }*d*. Thus, power density as a function of diameter can be expressed as the power density of a 1 mm beam divided by

^{2}*d*:

^{2}

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! )

Information source: technical note from Newport corporations Website

*You might also like to read: How to measure the peak power of a pulsed laser? *

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On Dec 18, 2012, Ritchie Buschow said:Reference to your article “A Shorcut for Calculating Power Density of a laser beam”. The fromula derivation may not be completely correct. When you solve for PowerDensity = Power(W) / πr2 = Power / π(0.5mm)2 ≈ 127 x Power (W/cm2); the answer should 1.27 x Power (W/cm2) unless I am missing something. Thanks

On Dec 20, 2012, admin said:Hi Ritchie,

Thanks for your comment. You’re correct that 1/(π*.5^2) = 4/π, or about 1.27. What you might not have noticed is that we are also converting mm^-2 to cm^-2, so we need to add two orders of magnitude. Thanks for pointing this out, as it may have been unclear to others as well.

On Jul 18, 2013, Leti said:hi,

what would be the formula to calculate the powerd density from a laser beam,if we also want to take into account the objective (40x) and magnification (1.6x)?

On Jul 28, 2013, admin said:I assume we are talking about using a micro objective to focus a laser beam. Calculating the power density at the focal spot is possible, but there is not a simple formula for it. You need to know the beam size as it enters the objective and the focal length of the objective. From these you can calculate the divergence (actually convergence) angle by common trigonometry. If the beam is Gaussian then the formulas relating beam size at the waist and divergence angle are given in the article:

http://bit.ly/1792lsX