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I am looking for a simple way of explaining the collimation of a laser beam. The typical discussion of the two slit experiment of quantum theory relies heavily on the Huygens principle. Its application to a laser beam would tend to predict spreading. From a purely electromagnetic field point of view, how can one visualize what happens at the edge of the beam?

Qmechanic
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2 Answers2

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Just as Jon Custer wrote in his comment, even a perfectly collimated laser beam with a planar wavefront will diverge. The way it happens is determined by the Huygens principle, and depends on the beam profile:

When the light intensity is abruptly cut by a sharp flat obstacle, the light will indeed diffract in almost all angles. A razor blade cutting a laser beam, even though viewed from the geometric shadow, will still have a bright rim. (In wave optics, no absolute shadows exist!)

When the intensity of the beam is modulated smoothly, it will diffract in small angles only. Intuitively it can be imagined that at the edge of a beam passing through a "soft aperture", the Huygens elementary source with slightly higher amplitude than its neighbour forces the resulting wavefront to be bent away by a minute angle, outwards from the beam axis. It is still all about the superposition of spherical waves.

Edit: To illustrate this, I employed my open-source simulation scripts, using the excellent MEEP Maxwell-equations solver, to a classical edge-diffraction experiment. In the following three animations, I computed for you a wave diffraction on a sharp edge (left), and on "soft apertures" with characteristic transition width of 50 % and 100 % of the full image width.

sharp-edge diffraction50% soft edge diffraction100% soft edge diffraction

dominecf
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  • Could you add a sketch or some images, please. – HolgerFiedler Feb 02 '16 at 17:59
  • @HolgerFiedler for some images and animations see e.g. wikipedia page on diffraction – Ruslan Feb 02 '16 at 19:24
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    Consider the "perfectly collimated" beam in your first paragraph. I am trying to visualize what happens at the edges. Or is that a mere mathematical abstraction of a beam whose edges smoothly go to zero as one proceeds further from the center of the beam? –  Feb 02 '16 at 19:49
  • @Heaviside Maybe this is not the most elegant physical approach, but you may envisage that, in compliance with Huygens principle, the center of the beam radiates into the surrounding area at circumference of the beam. The wave radiating diagonally is however retarded in time. Since the center of the beam is more intense, it partially imparts this phase retardation also to the resulting superposed wave. Its wavefront is then curved and the beam diffracts. -- This is the best way how I can envisage it without starting talking about spatial frequency and Fourier transform. – dominecf Feb 02 '16 at 20:03
  • @Ruslan It seems, that dominecf 's edit is much mor better than the link to diffraction on Wikipedia ;-) – HolgerFiedler Feb 02 '16 at 20:09
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    Only one additional issue: In discussions of relativity, shouldn't one restrict oneself to spherical wavefronts in derivations? It seems to me that the usual practice of assigning a single velocity (albeit of speed $c$) to a light "beam" is incorrect. A real "beam" consists of many photons going in slightly different directions. –  Feb 03 '16 at 03:30
  • @Heaviside You are absolutely true; it is always tricky to assign a velocity to a spatially limited beam, a short optical pulse propagating in a dispersive medium, or in a waveguide. Often you must discuss the phase velocity (that of the wavefront) and group velocity (that of the energy). The discussion of the information propagation velocity can be shown to be even more troublesome. In a focused beam, the phase velocity in its waist is even faster than speed of light; this is known as Gouy shift and it does not contradict relativity. – dominecf Feb 03 '16 at 08:04
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A properly collimated laser beam is called a Gaussian Beam whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function.

The Gaussian beam is a transverse electromagnetic (TEM) mode. The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation:

enter image description here

The width of such laser beam is described by:

enter image description here

The relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes Fraunhofer diffraction.

Here is how a Gaussian beam1 looks:

enter image description here

1. Excerpt from the following article: Quantitative and Qualitative Study of Gaussian Beam Visualization Techniques

Ziezi
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    Does a Gaussian beam have the least angular spread, in some sense? – DanielSank Feb 02 '16 at 20:19
  • @DanielSank Yes, such a beam can be focused into the "smallest" spot. – Ziezi Feb 02 '16 at 20:22
  • Any chance I can get you to define "smallest spot"? :) – DanielSank Feb 02 '16 at 20:30
  • @DanielSank "smallest spot" == most concentrated spot – Ziezi Feb 02 '16 at 20:36
  • @DanielSank The "smallest spot" is reached when illuminating the point from the largest possible solid angle (used in 4pi microscopes). What is interesting on the gaussian beam over all possible beam shapes, to write it simply, it reaches the smallest product of its original width and focused width, no matter how wide it is. – dominecf Feb 03 '16 at 08:08
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    @dominecf surely this is related to the fact that the Gaussian minimizes the product of the variance of the function and the variance of its Fourier transform? – DanielSank Feb 03 '16 at 18:12
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    @DanielSank Exactly as you write. It is also tightly connected with the concept of "minimum wavepacket" in quantum mechanics. In fact, focusing a beam onto the focal plane of a lens is equivalent to Fourier-transforming the beam profile. More generally, at any point between an ideal lens and the focal plane, the beam is determined by the fractional Fourier transform of its original profile. – dominecf Feb 03 '16 at 19:13
  • @dominecf Yeah this all makes sense. I had to re-load Fourier optics from long term storage :D Thanks very much. – DanielSank Feb 03 '16 at 19:23