page_title: Gaussian beam
section_title: Mathematical form
hierarchical_section_title: Gaussian beam / Mathematical form
caption_attribution_description: English: Surface plot of the intensity profile of a Gaussian laser beam with beam waist w 0 = 2 &#x03BB; {\displaystyle w_{0}=2\lambda } . The height profile is proportional to the beam intensity, while the color is proportional to the electromagnetic field amplitude. The white lines represent the beam width, both at the FWHM and 1/e² intensity level.
context_page_description: In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity profile. This fundamental transverse Gaussian mode describes the intended output of most lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse phase dependence is altered; this results in a different Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam are determined by a single parameter: the so-called waist w₀. At any position z relative to the waist along a beam having a specified w₀, the field amplitudes and phases are thereby determined as detailed below.
  The equations below assume a beam with a circular cross-section at all values of z; this can be seen by noting that a single transverse dimension, r, appears.
context_section_description: The Gaussian beam is a transverse electromagnetic (TEM) mode. The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation. Assuming polarization in the x direction and propagation in the +z direction, the electric field in phasor (complex) notation is given by:
  where
  r is the radial distance from the center axis of the beam,
  z is the axial distance from the beam's focus (or "waist"),
  i is the imaginary unit,
  k = 2πn/λ is the wave number (in radians per meter) for a free-space wavelength λ, and n is the index of refraction of the medium in which the beam propagates,
  E₀ = E(0, 0), the electric field amplitude (and phase) at the origin at time 0,
  w(z) is the radius at which the field amplitudes fall to 1/e of their axial values (i.e., where the intensity values fall to 1/e² of their axial values), at the plane z along the beam,
  w₀ = w(0) is the waist radius,
  R(z) is the radius of curvature of the beam's wavefronts at z, and
  ψ(z) is the Gouy phase at z, an extra phase term beyond that attributable to the phase velocity of light.
  There is also an understood time dependence eiωt multiplying such phasor quantities; the actual field at a point in time and space is given by the real part of that complex quantity.
  Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. In most practical cases (where w₀ ≫ λ/n) the above form is valid. Then the wave's associated magnetic field is everywhere directly proportional to the electric field and perpendicular to it, a direct consequence of Faraday's Law. Since the electric field is taken to be polarized in the x direction, the magnetic field is polarized in the y direction according to:
  where the constant η is the wave impedance of the medium in which the beam is propagating. For free space, η = η₀ ≈ 377 Ω.
  The intensity (or irradiance) distribution can then be found by evaluating the Poynting vector, which is entirely in the z direction:
  where I₀ = |E₀|²/2η is the intensity at the center of the beam at its waist and the superscript * symbolizes the complex conjugate. 
  If P₀ is the total power of the beam,
caption: Gaussian beam profile with w0 = 2λ
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