The beam in the cavity and the output beam of the laser, when travelling in free space ( or a homogenous medium ) rather than waveguides ( as in an optical fiber laser ), can be approximated as a Gaussian beam in most lasers ; such beams exhibit the minimum divergence for a given diameter.

It is also concerned with the physics of laser beam propagation, particularly the physics of Gaussian beams, with laser applications, and with associated fields such as nonlinear optics and quantum optics.

Gaussian beam propagation is a simple paraxial physical optics model for the propagation of coherent radiation such as laser beams.

The 1 / e < sup > 2 </ sup > width is important in the mathematics of Gaussian beams.

The divergence of good-quality laser beams is modeled using the mathematics of Gaussian beams.

Gaussian laser beams are said to be diffraction limited when their radial beam divergence is close to the minimum possible value, which is given by

Many lasers emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or " TEM < sub > 00 </ sub > mode " of the laser's optical resonator.

: is the Gouy phase shift, an extra contribution to the phase that is seen in Gaussian beams.

Non-Gaussian beams also exhibit this effect, but a Gaussian beam is a special case where the product of width and divergence is the smallest possible.

From the above expression for divergence, this means the Gaussian beam model is valid only for beams with waists larger than about.

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series ; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus ; in numerical analysis as Gaussian quadrature ; in finite element methods as Shape Functions for beams ; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.

The majority of optical tweezers make use of conventional TEM < sub > 00 </ sub > Gaussian beams.

However a number of other beam types have been used to trap particles, including high order laser beams i. e. Hermite Gaussian beam ( TEM < sub > xy </ sub >), Laguerre-Gaussian ( LG ) beams ( TEM < sub > pl </ sub >) and Bessel beams.

As a result, usually reference beams are Gaussian beams or spherical wave beams ( beams that radiate from a single point ) which are fairly easy to reproduce.

This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves ( light ) in the form of either paraboloidal waves or Gaussian beams.

* Gaussian integers: those complex numbers where both and are integers are also quadratic integers.

In 1976 the International Astronomical Union ( IAU ) revised the definition of the AU for greater precision, defining it as that length for which the Gaussian gravitational constant ( k ) takes the value when the units of measurement are the astronomical units of length, mass and time.

The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis ( the symmetrical axis of the system ) are infinitely small, i. e. with infinitesimal objects, images and lenses ; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.

Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit " sub-systems ", including Gaussian, " ESU ", " EMU ", and Heaviside – Lorentz.

Among these choices, Gaussian units are the most common today, and in fact the phrase " CGS units " is often used to refer specifically to CGS-Gaussian units.

** Normal dynamics, is a stochastic motion having a Gaussian probability density function in position with variance MSD that follows, MSD ~ t, where MSD is the mean squared displacement of the process, and t is the time the process is seen ( normal dynamics and Brownian dynamics are very similar ; the term used depends on the field )

This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers.

For general matrices, Gaussian elimination is usually considered to be stable in practice if you use partial pivoting as described below, even though there are examples for which it is unstable.

Gaussian elimination does not generalize in any simple way to higher order tensors ( matrices are order 2 tensors ); even computing the rank of a tensor of order greater than 2 is a difficult problem.

Many media transforms, such as Gaussian blur, are, like lossy compression, irreversible: the original signal cannot be reconstructed from the transformed signal.

Some terms associated with gravitational mass and its effects are the Gaussian gravitational constant, the standard gravitational parameter and the Schwarzschild radius.

Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

Other units commonly used are Gaussian units ( based on the cgs system ), Lorentz – Heaviside units ( used mainly in particle physics ) and Planck units ( used in theoretical physics ).

Methods that employ a distance function, such as nearest neighbor methods and support vector machines with Gaussian kernels, are particularly sensitive to this.

This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.

The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.

An example of such a domain is the Gaussian integers Z, that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers.

Its prime elements are known as Gaussian primes.

Rational primes ( i. e. prime elements in Z ) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.

An effective alternative is the singular value decomposition ( SVD ), but there are other less expensive choices, such as QR decomposition with pivoting ( so-called rank-revealing QR factorization ), which are still more numerically robust than Gaussian elimination.

The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series.

The above Gaussian wavepacket, unnormalized and just centered at the origin, instead, can now be written in 3D:

In general, a smoothing filter sets each pixel to the average value, or a weighted average, of itself and its nearby neighbors ; the Gaussian filter is just one possible set of weights.

It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units.

The term " cgs units " is ambiguous and therefore to be avoided if possible: cgs contains within it several conflicting sets of electromagnetism units, not just Gaussian units, as described below.

The Gaussian unit system is just one of several electromagnetic unit systems within CGS.

In SI, 1 / ε < sub > 0 </ sub >, converts or scales flux density, D, to electric field, E ( the latter has dimension of force per charge ), while in rationalized Gaussian units, flux density is the very same as electric field in free space, not just a scaled copy.

Here, just as in Gaussian elimination, there are additional parameters which appear in attempting to absorb the torsion.