How to Find the Waist of a Gaussian Beam at the Focus Using a Lens?

[Nanxie]

Homework Statement

Hello, I'm really confused about this one, =(
Suppose I have a guassian beam that is colliminated (beam expanded) and then suppose i let it passed through a lens to focus it.

Given an initial waist and focal length of the lens. How can i find the waist (or half width) at the focus?

Homework Equations

The Attempt at a Solution

There's an approximation using the formula above

w(z)=(lambda*z)/(pi*w0)

where w0 is the waist at focus and z is the distance from it, so if i let z=f i can solve for w0
but this only works if z >> pi(w0)/lambda

how about for the case when z is small (or the focal length is small), i can't seem to solve it using the above equations.

Thanks~ ^^;

 

[Redbelly98]

Welcome to PF!

Sorry about the late reply, don't know how I missed your question earlier.

There's another equation for the angle of a Gaussian beam, given it's wavelength and waist size (at the focus). You can use that, as long as the focal length is much much longer than Z_R -- and you can verify if that's true after you have found w_o at the focus.

EDIT: Oh, I see you want a more general result, one that does not assume f>>Z_R . Hang on while I check in my old Lasers textbook ... I'll post back soon

EDIT #2: Okay. What you want is to look up the "q parameter" for Gaussian beams, and how ABCD matrices can be used in calculating Gaussian beam propagation. Do you have a textbook that discusses these in detail?

 

[Nanxie]

looked up the 'q parameter' and you're right~
Just need the Matrix ABCD to calculate the beam parameters at any z.

thank you! ^^