How Is Laser Beam Intensity Calculated?

[ganondorf29]

Homework Statement

A helium-neon laser of the type often found in physics laboratories has a beam power of 3.50 mW at a wavelength of 633 nm. The beam is focused by a lens to a circular spot whose effective diameter may be taken to be equal to 2.00 wavelengths. Calculate the intensity of the focused beam.

Homework Equations

$$I = P / 4\pi r^2$$

The Attempt at a Solution

$$I = P / 4\pi r^2$$

P = 3.50mW
I don't know what the radius is. From the problem, I used 633nm as the radius because the diameter = 2*(633nm), I used 633nm as the radius, but after using that value the answer is wrong. If someone can tell me the radius, I think I could get this problem. Thanks

 

[tiny-tim]

Hi ganondorf29!

(have a pi: π )

A helium-neon laser of the type often found in physics laboratories has a beam power of 3.50 mW at a wavelength of 633 nm. The beam is focused by a lens to a circular spot whose effective diameter may be taken to be equal to 2.00 wavelengths. Calculate the intensity of the focused beam.
…
I = P / 4\pi r^2

No, that's the wrong formula …

as you can see, it's the total power divided by the surface area of a whole sphere of radius r …

it applies when the beam is all-round, so the intensity is P/4πr² , and the power through an area A (at distance r) is PA/4πr².

In this case, the total power is confined to the beam, and it all goes through the "effective circle", so the appropriate formula for intensity is … ?

 

[nlightner]

!

The radius in this case would be half of the diameter, so it would be 633nm/2 = 316.5nm.

Therefore, the intensity of the focused beam would be:

I = 3.50mW / (4 * pi * (316.5nm)^2) = 1.23 mW/nm^2

This is a relatively high intensity, which is expected for a laser beam. It is important to note that this calculation assumes that the beam is perfectly focused and there is no loss of power due to scattering or other factors. In a real laboratory setting, the actual intensity may be slightly lower.