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Gaussian beams and paraxial approximation



Gaussian beams  
is a beautiful phenomenon in optics. As they propagate through space they retain their Gaussian shape, and only get broader or narrower. They are symmetric along the optical axis. No matter how many lenses you use to focus and defocus your laser beam, it will remain Gaussian. And it's shape is described by a few simple formulas, which define their thinnest section (w0, 'waist'), radius of wavefront (R), and divergence angle (Theta). Some immediate applications include fiber coupling, confocal microscopy, and light-sheet microscopy.

The formulas describing Gaussian beams were derived in the 1960-s, soon after the invention of lasers, by solving the wave equation for electromagnetic waves, and were analysed exhaustively in paraxial approximation (Kogelnik and Li, 1966).

Paraxial approximation means that angle of beam divergence angle is small (theta ~ tan(theta)). However, modern microscopy pushes limits to high-NA objectives and laser beams for higher resolution.

How good is the paraxial approximation? What if you focus a Gaussian beam to a really small spot of an order of a wavelength? Do these formulas still apply?
The short answer is yes, they apply quite well. As shown in (Agrawal and Pattanayak, 1979), non-paraxial description adds a lot of math, but not so much difference in the beam shape:
The beam profiles shown in (a) are cross-sections simulated about 1 wavelength from the waist (really close), with effective NA(air) about 1.0. The difference between paraxial and exact formulas is mostly over-estimation of peak intensity in paraxial case. The side lobes, and hence the width of beams, look remarkably similar. As you go farther from the waist, the paraxial approximation gets more accurate, too.

So if you are doing microscopy with high-NA Gaussian beams, stay cool - the old formulas from textbooks are good.

Here is an excellent review on Gaussian beams in light-sheet microscopy (Power & Huisken, Nature Methods, 2017)

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