Laser Beam - Divergence and Solid Angle

[Naz93]

I am somewhat confused about the connection between divergence and solid angle for a beam. I know individually what each term means... but I'm confused as to how (or even if) one can calculate the solid angle of a beam, given the divergence.

I have some notes from a previous lecture series I attended on laser physics, where I wrote down: "Laser radiation has a narrow divergence (~1mrad), and so a small solid angle (~10^-6∏ sterad)". That seems to imply that the divergence is sufficient information to calculate the solid angle of the beam... but I just can't seem to do it.

(I'm not concerned just with the numbers given here, but rather the general problem of how to relate solid angle and beam divergence)

Thanks!

 

[rigetFrog]

Are you looking for an equation? Or do you need help understanding an existing equation?

 

[Naz93]

I don't know of an equation to calculate solid angle from beam divergence... I only know solid angle by the definition dΩ=sinθ*dθ*dø

I feel like I'm missing something obvious here...

 

[rigetFrog]

Is this what you're looking for?

http://en.wikipedia.org/wiki/Diffraction#Circular_aperture

If not, could you elaborate on your question some more. Maybe draw a picture and put a question mark on the part you don't understand.

 

[Naz93]

Hmm, that kind of explains it. All I'm really after is a way to calculate a number for the solid angle of a beam, given a number for the divergence of the beam... I'm still not sure how to do this

 

[rigetFrog]

the equation above looks correct. Are you looking for theta or phi? Or are you trying to figure out how to solve the eq above?

 

[Naz93]

Well... I don't know what theta or phi even are in relation to the beam divergence. The equation is a correct expression for a solid angle in terms of the two polar angle coordinates, but I don't see how it helps in finding the solid angle given a beam divergence...

 

[rigetFrog]

Ok this should answer your question

http://en.wikipedia.org/wiki/Beam_divergence

 

[UltrafastPED]

Divergence angle is used because it is relatively easy to measure given that the beam has a circular profile. The formulas given in the article are for the ideal Gaussian beam, which is often, but not always what is output from a laser.

For the simplest application just determine the diameter of the circle which contains 90% of the beam power (or your favorite proxy for beam width) at two points sufficiently far apart. Then apply the formula pointed out by rigetFrog.

 

[Naz93]

Thanks both. The formula would then give the divergence, calculated from the two beam diameters and the distance between them... but then I'm still confused how to get the solid angle of the beam from the divergence. I think I'm just confused what a solid angle even is...

 

[UltrafastPED]

Thanks both. The formula would then give the divergence, calculated from the two beam diameters and the distance between them... but then I'm still confused how to get the solid angle of the beam from the divergence. I think I'm just confused what a solid angle even is...

The cone of revolution about the central ray of the beam, whose angle is the divergence angle, will intersect the surface of every sphere whose center is the apex of the cone.

The solid angle is the ratio of the surface area enclosed by the cone to the total surface area of the sphere. Pick the distance R=1 as being the most convenient surface.

Solid angle: https://en.wikipedia.org/wiki/Solid_angle

 

[Naz93]

But how to project a circle onto the curved surface of a sphere? Is the way to actually calculate a solid angle to use the equation on wikepedia (integral of sin(theta)d(theta)d(phi) ) with the limits of theta and phi both set to the divergence angle?

 

[UltrafastPED]

See eq. (2) here: http://www.physicsinsights.org/solid_angles_1.html

The author provides a good definition of the solid angle.

 

[Naz93]

Amazing, thanks! That's a really nice explanation :-)