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In the case of $TEM_{00}$ mode laser produces gaussian beam. I read in wikipedia that it converges till some point called 'beam waist'and then it diverges to infinity.

If there is a lens placed along the beam, then it converges to the focal point of lens and then it diverges. Here the beam can't converge to a single point but has a non-zero width.

  1. Is it because of the Hesienberg uncertainty principle that $\Delta x$ becomes zero which further implies $\Delta p$ tends to infinity? And the momentum can't be infinite, so it doesn't get focus to a single point. Am I right?

  2. I read that it happens even without a lens. How does focusing of beam happen without lens?

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  • $\begingroup$ for your point 2: I think you can have a focusing gaussian beam in the output of a cavity formed by 2 curved mirrors with the same sign of curvature : something like ((> . Somehow equivalent to say that the "lens" is within the cavity. Many lasers have the waist at the output face (with a plano-curve cavity). $\endgroup$ – scrx2 Nov 21 '15 at 21:13
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On point 1, you are correct. The uncertainty principle leads to the finite spot size when focusing a laser beam and the spread of the beam thereafter. In general, in order to produce a smaller spot, you need to focus the beam in at a high angle (short focal length lens), meaning a larger spread in transverse momentum of the photons. The same thing can be said for imaging optics. In order for a microscope to look at a small object, you need a short focal length microscope objective that can collect light at a high angular spread.

As for point 2, "focusing" of a laser beam without a lens can happen due to some non-linear effects if you are at very-high intensities in some type of medium (air or otherwise), but that is a different subject all together. However, spreading of a laser beam is an inherent property of the beam due to the uncertainty relationship as well. Because the laser has a finite spot size, it must also have a spread in the transverse momentum. This means that as the laser propagates, the beam will in general expand. The far-field angular divergence (full angle of the "cone" of the beam) is given by (for $TEM_{0,0}$):

$\theta = \frac{2\lambda_0}{\pi n w_0}$

where $\lambda_0$ is the wavelength, $n$ is the index of refraction and $w_0$ is the waist size of the beam. You can see here that the divergence angle is inversely proportional to the waist size of the beam, $w_0$. For larger beams, there is less divergence.

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  • $\begingroup$ In the wikipedia article,en.wikipedia.org/wiki/Gaussian_beam,it is written that laser beam is gaussian and that it is described by a spot radius. But this is given independent of using lens. Is it saying that laser beam when it comes out of cavity initially converges and later diverges. Please make this clear to me. $\endgroup$ – Pavan Oct 16 '15 at 10:47
  • $\begingroup$ The spot size of the beam is an inherent property of a Gaussian beam, and doesn't indicate that the beam has to focus coming out of the laser cavity if a lens is not present. The spot size itself for a Gaussian beam is directly related to its divergence angle. In some cases, the location of the minimum spot size can be virtual, meaning that it is actually "behind" the laser output and is never achieved by the actual laser output. The spot size allows you to calculate how the beam size changes from the output if you have a few measurements of the beam profile after the laser output. $\endgroup$ – tmwilson26 Oct 16 '15 at 11:07

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