What Is the Power Ratio in Gaussian Beams Through a Circular Aperture?

[thewall12]

Homework Statement

Determine the ratio of the power contained within a circle of radius of w0 (in the transverse plane at z = 0) to the total power in the Gaussian beams of orders (0,0), (1,0), (0,1) and (1,1). Then compare the ratio of the power contained within a circle of radius w0/3 to the total power for the (0,0) and (1,1) Gaussian beams. (This transmission difference is the principle behind spatial filtering that is used to select the principal mode from a multimode beam.) You may find it easier to use cylindrical coordinates for this problem.

Homework Equations

The Attempt at a Solution

I've attached the solutions. The solutions are for #4. I'm not exactly sure how power and electric field are proportional. Can anybody explain that? Also, I don't understand how the integrals for TEM 01/10 and TEM11 were formulated.

 

[L-x]

Do you know the formula for the intensity distribution of a gaussian beam? (remind yourself of it here: http://en.wikipedia.org/wiki/Gaussian_beam)

Notice (also confirmed on that page) that intensity is proportional to power and you should be able to convince yourself that the relationship quoted in the solutions is indeed correct.

It may be helpful to post your own attempts at formulating the integrals you mentioned, so people can help you understand them.

 

[thewall12]

Thanks, L-x. I can see the relationship now. I'm still a little lost on the integrals, though. I know that since the beam's optical axis is z, the cylindrical coordinates (r, theta, z) just become a function of r and theta. I understand the bounds on the integrals and the E^2 portion, but not how you have r, r^3, and r^5.