Gaussian beam
This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other.
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.
[2] The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation.
The time factor involves an arbitrary sign convention, as discussed at Mathematical descriptions of opacity § Complex conjugate ambiguity.
Although there are other modal decompositions, Gaussians are useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis.
This includes the Rayleigh range zR and asymptotic beam divergence θ, as detailed below.
[8] Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where r = w(z).
The total angular spread of the diverging beam, or apex angle of the above-described cone, is then given by
That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large numerical aperture, in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength.
[10] With a beam centered on an aperture, the power P passing through a circle of radius r in the transverse plane at position z is[11]
The reciprocal of q(z) contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:[12]
As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the phase that is added to each point
[15] An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.
is This last expression makes clear that the ray optics thin lens equation is recovered in the limit that
In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot.
As a special case of electromagnetic radiation, Gaussian beams (and the higher-order Gaussian modes detailed below) are solutions to the wave equation for an electromagnetic field in free space or in a homogeneous dielectric medium,[17] obtained by combining Maxwell's equations for the curl of E and the curl of H, resulting in:
where c is the speed of light in the medium, and U could either refer to the electric or magnetic field vector, as any specific solution for either determines the other.
The Gaussian beam solution is valid only in the paraxial approximation, that is, where wave propagation is limited to directions within a small angle of an axis.
The fundamental Gaussian happens to be the one that minimizes the product of minimum spot size and far-field divergence, as noted above.
Two important orthogonal decompositions of this sort are the Hermite–Gaussian or Laguerre-Gaussian modes, corresponding to rectangular and circular symmetry respectively, as detailed in the next section.
Such a solution is possible due to the separability in x and y in the paraxial Helmholtz equation as written in Cartesian coordinates.
The fourth factor is the Hermite polynomial of order J ("physicists' form", i.e. H1(x) = 2x), while the fifth accounts for the Gaussian amplitude fall-off exp(−x2/w(z)2), although this isn't obvious using the complex q in the exponent.
Expansion of that exponential also produces a phase factor in x which accounts for the wavefront curvature (1/R(z)) at z along the beam.
Hermite-Gaussian modes are typically designated "TEMlm"; the fundamental Gaussian beam may thus be referred to as TEM00 (where TEM is transverse electro-magnetic).
The only specific difference in the x and y profiles at any z are due to the Hermite polynomial factors for the order numbers l and m. However, there is a change in the evolution of the modes' Gouy phase over z:
On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.
As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in r but now multiplied by a Laguerre polynomial.
The effect of the rotational mode number l, in addition to affecting the Laguerre polynomial, is mainly contained in the phase factor exp(−ilφ), in which the beam profile is advanced (or retarded) by l complete 2π phases in one rotation around the beam (in φ).
This is an example of an optical vortex of topological charge l, and can be associated with the orbital angular momentum of light in that mode.
[7] There is another important class of paraxial wave modes in cylindrical coordinates in which the complex amplitude is proportional to a confluent hypergeometric function.
