Gaussian beam

In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse gaussian mode describes the intended output of most (but not all) 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 (for a given wavelength and polarization) are determined by a single parameter: the so-called waist w0. At any position z relative to the waist (focus) along a beam having a specified w0, the field amplitudes and phases are thereby determined as detailed below. In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse gaussian mode describes the intended output of most (but not all) 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 (for a given wavelength and polarization) are determined by a single parameter: the so-called waist w0. At any position z relative to the waist (focus) 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. Beams with elliptical cross-sections, or with waists at different positions in z for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of w0 and of the z = 0 location for the two transverse dimensions x and y. Arbitrary solutions of the paraxial Helmholtz equation can be expressed as combinations of Hermite–Gaussian modes (whose amplitude profiles are separable in x and y using Cartesian coordinates) or similarly as combinations of Laguerre–Gaussian modes (whose amplitude profiles are separable in r and θ using cylindrical coordinates). At any point along the beam z these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different Gouy phase which is why the net transverse profile due to a superposition of modes evolves in z, whereas the propagation of any single Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam. Although there are other possible modal decompositions, these families of solutions are the most useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is not operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). 'Gaussian beam' normally implies radiation confined to the fundamental (TEM00) Gaussian mode. 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 There is also an understood time dependence e i ω t {displaystyle e^{iomega t}} 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) the above form is valid. Then the wave's associated magnetic field is everywhere directly proportional to the electric field and perpendicular to it. 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.