Understanding Gaussian Beam Contraction and Divergence in Optics

[zebanaqvi]

I am new to lasers. In the expression for q parameter, 1/q = 1/R - j(λ/πw^2)
how did we come to know that w(z) is a measure of decrease in field amplitude E with distance? I can't feel it.

Does the gaussian beam itself contract to the minimum diameter? Shouldn't a lens be required for this? I can understand the divergence of the beam but not the contraction.

Can anybody please make me understand both mathematically and physically?
Regards,
Zeb

 

[wukunlin]

if you "start" with a collimated beam, then it will always expand in diameter.

but sometimes for a variety of reasons you can get a beam that will have beams with decreasing diameter as it propagates until it forms the beams waist and start to expand again.
Such beams are easily created, like you said, with lenses. fabry-perot laser beams sometimes come with a decreasing diameter probably due to the non linearity of the components in the resonator.

does that answer your question?

 

[zebanaqvi]

Thanks a lot :)
Sorry, what I am about to ask might be lame.
I have the helmholtz equation, and I want to solution to represent a beam whose phase front is not plane always, whose intensity profile is not uniform across the cross section. What function should I try out? You'll say complex. Something like psi(x,y,z)exp(-jkz) where psi is a complex function. Please explain me why should psi be complex? Is it that only a complex function will be able to represent the desired beam? y?

 

[wukunlin]

in most cases in optics psi is easier to work with if it is complex, there are more discussion on this in this thread:
https://www.physicsforums.com/showthread.php?t=60069