Measuring Gaussian Beams

From Qwiki

The parameters for a Gaussian beam can be determined from two measurements of the beam diameter separated by a known distance $x$ in conjunction with the knowledge of whether the lies between or outside of the two measurements. The only knowledge of Gaussian beam propagation needed is the beam waist equation

$w^2(z) = \frac{\lambda z_R }{\pi}\left(1+\left(\frac{z}{z_R}\right)^2\right).$

$w 2 (z)$ is the beam diameter at $z$ (called $w 1$ from now on) and $z R$ is the Raleigh range, which we will solve for.

Let $z$ be the location of the first measurement, and $z + x$ be the location of the second. The second measurement is related to the Raleigh range by

$w^2(z+x)\equiv w_2^2 = \frac{\lambda z_R }{\pi}\left(1+\left(\frac{z+x}{z_R}\right)^2\right).$

Solving the first equation for $z 2$ ( $z^2= w_1^2 z_R \pi/\lambda -z_R^2$ ) and then subtracting the first equation from the second and solving this for $z R$ gives us a quadratic equation for $z$ ,

$\left(w_2^2-w_1^2+\frac{\lambda^2}{\pi^2}\frac{4x^2}{w_2^2-w_1^2}\right)z^2 + \left(\frac{\lambda^2}{\pi^2}\frac{4x^3}{w_2^2-w_1^2} -2w_1^2x\right)z + \left(\frac{\lambda^2}{\pi^2}\frac{x^4}{w_2^2-w_1^2} -w_1^2x^2\right) =0$

There are two solutions to the quadratic equation, the plus sign gives the solution for the beam waist outside of the measurements and the minus is the solution when the waist is between the measurements. In order for the measurements to be consistent with a real beam, the radicand must be positive,