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The Gaussian function has a 1/e 2 diameter (2w as used in the text) about 1.7 times the FWHM.. At a position z along the beam (measured from the focus), the spot size parameter w is given by a hyperbolic relation: [1] = + (), where [1] = is called the Rayleigh range as further discussed below, and is the refractive index of the medium.
In optics, the complex beam parameter is a complex number that specifies the properties of a Gaussian beam at a particular point z along the axis of the beam. It is usually denoted by q . It can be calculated from the beam's vacuum wavelength λ 0 , the radius of curvature R of the phase front , the index of refraction n ( n =1 for air), and ...
The Gaussian curvature is the product of the two principal curvatures Κ = κ 1 κ 2. The sign of the Gaussian curvature can be used to characterise the surface. If both principal curvatures are of the same sign: κ 1 κ 2 > 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface ...
If we have a Gaussian beam of wavelength , radius of curvature R (positive for diverging, negative for converging), beam spot size w and refractive index n, it is possible to define a complex beam parameter q by: [8] =.
The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. In particular the Gaussian curvature is an invariant of the metric, Gauss's celebrated Theorema Egregium. A convenient way to understand the curvature comes from an ordinary differential equation, first ...
Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory. Gaussian beams are used in optical systems, microwave systems and lasers. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing.
Gaussian beam width () as a function of the axial distance .: beam waist; : confocal parameter; : Rayleigh length; : total angular spread In optics and especially laser science, the Rayleigh length or Rayleigh range, , is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. [1]
Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.