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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 ...
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] =.
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]
If the beam is distributed in phase space with a Gaussian distribution, the emittance of the beam may be specified in terms of the root mean square value of and the fraction of the beam to be included in the emittance. The equation for the emittance of a Gaussian beam is: [1]: 83
For a Gaussian beam, no simple upper integration limits exist because it theoretically extends to infinity. At r >> R, a Gaussian beam and a top-hat beam of the same R and S 0 have comparable convolution results. Therefore, r ≤ r max − R can be used approximately for Gaussian beams as well.
A Gaussian beam has the lowest possible BPP, /, where is the wavelength of the light. [1] The ratio of the BPP of an actual beam to that of an ideal Gaussian beam at the same wavelength is denoted M 2 ("M squared"). This parameter is a wavelength-independent measure of beam quality.
This function is known as a super-Gaussian function and is often used for Gaussian beam formulation. [5] This function may also be expressed in terms of the full width at half maximum (FWHM), represented by w : f ( x ) = A exp ( − ln 2 ( 4 ( x − x 0 ) 2 w 2 ) P ) . {\displaystyle f(x)=A\exp \left(-\ln 2\left(4{\frac {(x-x_{0})^{2 ...