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

 

[PJC01]

As a scientist in the field of optics, I can certainly provide some insight into Gaussian beam contraction and divergence. The q parameter, 1/q = 1/R - j(λ/πw^2), is a representation of the complex beam parameter in Gaussian beam optics. This parameter is used to describe the size and shape of the Gaussian beam as it propagates through space.

In this expression, w(z) represents the beam waist, which is the minimum diameter of the beam. This parameter is related to the decrease in field amplitude with distance by the equation E(z) = E0 * exp(-z^2/w(z)^2), where E0 is the initial field amplitude and z is the distance from the beam waist. This means that as the beam propagates, the field amplitude decreases with distance from the beam waist, which is a measure of the beam's contraction.

It is important to note that the Gaussian beam itself does not physically contract to the minimum diameter. This minimum diameter is a mathematical representation of the beam's intensity distribution. In reality, the beam will continue to diverge due to diffraction effects. However, the Gaussian beam model is a useful approximation for describing the behavior of laser beams in many practical situations.

To understand this mathematically, we can look at the expression for the beam waist, w(z) = w0 * √(1 + (λ*z/π*w0^2)^2), where w0 is the initial beam waist and λ is the wavelength of the laser. This equation shows that as the beam propagates, the beam waist increases with distance, leading to a decrease in the field amplitude.

Physically, we can think of the beam waist as the point where the beam is most tightly focused. As the beam propagates away from this point, it begins to spread out due to diffraction, resulting in a decrease in the field amplitude. This is why a lens is often required to focus the beam to a desired spot size.

I hope this explanation helps you to better understand the concepts of Gaussian beam contraction and divergence in optics. Remember that this is a simplified model and that in reality, the beam will continue to diverge due to diffraction effects. However, the Gaussian beam model is a useful tool for describing the behavior of laser beams in many practical applications.