How to Evaluate Power Ratio in Higher Order Hermite-Gaussian Beams?

[semc]

Homework Statement

Evaluate the ratio of the power contained within a circle of radius W(z) in the transverse plane to the total power in the Hermite-Gaussian beams of order (1,0)

Homework Equations

P=$\int$IdA

The Attempt at a Solution

I have determined the ratio for the Gaussian beam but the higher order modes have a extra 'x' or 'y' in the integral but I am integrating w.r.t r (r²=x²+y²). So how can I evaluate the integral?

 

[ivanis]

Why don't you integrate in Decartes coordinates x and y? Because, if I understood correctly, then the integral is can be represented as
$\int^{W(z)}_{0} x*e^{-x^2}dx \int^{\sqrt{W(z)^2-x^2}}_{0}y*e^{-y^2}dy$
this is only a quarter of the full energy of course.

 

[semc]

I don't understand what is Decartes coordinates but I managed to solve it. Its simply x=rcos$\vartheta$. Have no idea why I didn't think of that when I posted this. Thanks