Gaussian beams are considered in conditions of small beam divergence, in order to apply paraxial approximation. A mathematical expression for the calculation of complex electric field amplitude of a monochromatic beam, propagating along the z-direction with wavelength λ, is given by:

Modes of high transverse order, on the other hand, are known as Laguerre-Gaussian, or Hermite-Gaussian, functions. Any deviation from a Gaussian beam shape can be estimated by the M2 factor. A Gaussian beam, having the highest beam quality, corresponds to M2 = 1.

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COMSOLGaussian beam

A Gaussian beam is a beam of electromagnetic radiation with intensity distributions, and a transverse electric field represented by Gaussian functions. Gaussian beams are named after the physicist and mathematician Johann Carl Friedrich Gauβ.

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Laguerre-Gaussian mode

Gaussian modes are the lowest order modes of an optical resonator, set along the transverse direction. A stable resonator will have homogeneous optical media, and a flat or parabolic surface between the media. Hence, lasers in the fundamental transverse mode often emit beams close to Gaussian shape.

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laguerre-gaussianbeam

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Reuven Silverman of Ophir discusses the critical role of M2 measurements in laser technology for optimization and quality control in various industries.

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Following refraction by a diffraction-limited lens, a Gaussian beam is converted into another beam with a different set of parameters. The mathematical function representing the Gaussian beam is a resultant of the Helmholtz equation in paraxial form. This function is used to calculate the complex amplitude of the electric field of the beam.

Besselbeam

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Mathematics of vectorialgaussianbeams

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where:      E0 = peak amplitude      w0 = beam radius      k = wave number = 2π/λ      zR = Rayleigh number      R(z) = radius of curvature of wavefronts.

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