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Nature of Light
Published in George K. Knopf, Kenji Uchino, Light Driven Micromachines, 2018
where the beam parameter product (BPP) is the product of a laser beam divergence and the diameter of the beam at its narrowest point (i.e., beam waist). The M2 beam quality factor provides a realistic representation of the propagation characteristics of the laser beam.
Paraxial Propagation of Gaussian Beams
Published in Glen D. Gillen, Katharina Gillen, Shekhar Guha, Light Propagation in Linear Optical Media, 2017
Glen D. Gillen, Katharina Gillen, Shekhar Guha
A commonly used, and rather handy, parameter to describe focused laser beams is the beam parameter product (also referred to as the beam propagation product), or BPP, which is the product of a beam’s minimum width ωo multiplied by the far-field divergence angle θo. For a TEM00 beam propagating through vacuum or air (n ≈ 1) the BPP is BPP=ωoθo=λπ.(6.26) BPP is usually expressed in units of mm mrad. Its usefulness comes from the fact that the product of these two parameters for a TEM00 beam of a given wavelength is constant. This direct relationship between the focal spot width and the convergence angle makes it straight forward to predict and manipulate the focal properties of a beam with a relatively high degree of accuracy. For example, if you desire a new focal spot to be twice as small as the original focal spot you will need twice the convergence angle. In other words, if one lens has half the focal length of another, then simply replacing the original lens with the shorter focal length lens would result in the desired doubling of the convergence angle, and the new beam would have a focal spot with a width half that of the original.
M2 factor of conically refracted Gaussian beams
Published in Journal of Modern Optics, 2022
Erko Jalviste, Viktor Palm, Viktor Peet
The factor is the ratio of the beam parameter product (BPP, the product of the waist radius and the divergence half-angle) of a beam under study to that of a Gaussian beam, for which it is equal to [5]. According to the ISO 11146 standard [6], the factors for two pre-selected transverse directions are defined by where , are beam diameters at the beam waist location (at focal plane) and are the divergence (full) angles, z is the longitudinal coordinate, and is the wave vector module. The σ notation refers to the second-moment root mean square (RMS) value of a parameter [32]. Equation (2) relates the divergence angles in x and y directions to the corresponding k-space radii and . Combining Equation (1) and (2) leads to For an axially symmetric beam, , and , thus it is reasonable to introduce RMS radii by defining In particular, for a Gaussian beam with waist radius , Combining Equations (3)–(5) results in When is defined in the spatial frequency () scale, like in Refs. [1,16,18], rather than in transverse wave vector (k) scale, the corresponding expressions (3) and (7) have an extra multiplier.